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BEGIN:VEVENT
SUMMARY:Andrew Granville (Université de Montréal)
DTSTART:20200430T150000Z
DTEND:20200430T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/1
DESCRIPTION:Title: Frobenius's postage stamp problem\, and beyond...\nby Andr
ew Granville (Université de Montréal) as part of Number Theory Web Semin
ar\n\n\nAbstract\nWe study this famous old problem from the modern perspec
tive of additive combinatorics\, and then look at generalizations.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Sutherland (MIT)
DTSTART:20200507T150000Z
DTEND:20200507T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/2
DESCRIPTION:Title: Sums of three cubes\nby Andrew Sutherland (MIT) as part of
Number Theory Web Seminar\n\n\nAbstract\nIn 1953 Mordell asked whether on
e can represent 3 as a sum of three cubes in any way other than $1^3+1^3+1
^3$ and $4^3+4^3 -5^3$. Mordell's question spurred many computational inve
stigations over the years\, and while none found a new solution for 3\, th
ey eventually determined which of the first 100 positive integers $k$ can
be represented as a sum of three cubes in all but one case: $k=42$.\n\nIn
this talk I will present joint work with Andrew Booker that used Charity E
ngine's crowd-sourced compute grid to affirmatively answer Mordell's quest
ion\, as well as settling the case $k=42$. I will also discuss a conjectur
e of Heath-Brown that predicts the existence of infinitely many more solut
ions and explains why they are so difficult to find.\n\nMSC:11Y50\, MSC:11
D25\, ACM:F.2.2\, ACM:G.2.3\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Blomer (Universität Bonn)
DTSTART:20200514T150000Z
DTEND:20200514T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/3
DESCRIPTION:Title: Joint equidistribution and fractional moments of L-functions\nby Valentin Blomer (Universität Bonn) as part of Number Theory Web Se
minar\n\n\nAbstract\nIntegral points on spheres of large radius $D^{1/2}$
equidstribute (subject to appropriate congruence conditions)\, and so do H
eegner points of large discriminant $D$ on the modular curve. Both sets ha
ve roughly the same cardinality\, and there is a natural way to associate
with each point on the sphere a Heegner point. Do these pairs equidstribut
e in the product space of the sphere and the modular curve as $D$ tends to
infinity?\n\nA seemingly very different\, but structurally similar joint
equidistribution problem can be asked for the supersingular reduction at t
wo different primes of elliptic curves with CM by an order of large discri
minant $D$.\n\nBoth equidistribution problems have been studied by ergodic
methods under certain conditions on $D$. I will explain how to use number
theory and families of high degree $L$-functions to obtain an effective e
quidistribution statement with a rate of convergence\, assuming GRH. This
is joint work in progress with F. Brumley.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michel Waldschmidt (Sorbonne University)
DTSTART:20200512T080000Z
DTEND:20200512T090000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/4
DESCRIPTION:Title: Representation of integers by cyclotomic binary forms\nby
Michel Waldschmidt (Sorbonne University) as part of Number Theory Web Semi
nar\n\n\nAbstract\nThe representation of positive integers as a sum of two
squares is a classical problem studied by Landau and Ramanujan. A similar
result has been obtained by Bernays for positive definite binary form. In
joint works with Claude Levesque and Etienne Fouvry\, we consider the rep
resentation of integers by the binary forms which are deduced from the cyc
lotomic polynomials. One main tool is a recent result of Stewart and Xiao
which generalizes the theorem of Bernays to binary forms of higher degree.
\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Felipe Voloch (University of Canterbury)
DTSTART:20200609T000000Z
DTEND:20200609T010000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/6
DESCRIPTION:Title: Value sets of sparse polynomials\nby Felipe Voloch (Univer
sity of Canterbury) as part of Number Theory Web Seminar\n\n\nAbstract\nWe
obtain a lower bound on the size of the value set $f(F_p)$ of a sparse po
lynomial $f(x)$ in $F_p[x]$ over a finite field of $p$ elements when $p$ i
s prime. This bound is uniform with respect to the degree and depends on t
he number of terms of $f$.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Timothy Browning (IST Austria)
DTSTART:20200604T150000Z
DTEND:20200604T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/7
DESCRIPTION:Title: Random Diophantine equations\nby Timothy Browning (IST Aus
tria) as part of Number Theory Web Seminar\n\n\nAbstract\nI’ll survey so
me of the key challenges around the solubility of polynomial Diophantine e
quations over the integers.\n\nWhile studying individual equations is ofte
n extraordinarily difficult\, the situation is more accessible if we merel
y ask what happens on average and if we restrict to the so-called Fano ran
ge\, where the number of variables exceeds the degree of the polynomial.
Indeed\, about 20 years ago\, it was conjectured by Poonen and Voloch that
random Fano hypersurfaces satisfy the Hasse principle\, which is the simp
lest necessary condition for solubility. After discussing related results
I’ll report on joint work with Pierre Le Boudec and Will Sawin where we
establish this conjecture for all Fano hypersurfaces\, except cubic surfa
ces.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Sarnak (IAS and Princeton University)
DTSTART:20200625T150000Z
DTEND:20200625T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/8
DESCRIPTION:Title: Integer points on affine cubic surfaces\nby Peter Sarnak (
IAS and Princeton University) as part of Number Theory Web Seminar\n\n\nAb
stract\nThe level sets of a cubic polynomial in four or more variables ten
ds to have many integer solutions\, while ones in two variables a limited
number of solutions. Very little is known in case of three variables. For
cubics which are character varieties (thus carrying a nonlinear group of m
orphisms) a Diophantine analysis has been developed and we will describe i
t. Passing from solutions in integers to integers in say a real quadratic
field there is a fundamental change which is closely connected to challeng
ing questions about one-commutators in $SL_2$ over such rings.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kristin Lauter (Microsoft Research Redmond Labs)
DTSTART:20200519T000000Z
DTEND:20200519T010000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/9
DESCRIPTION:Title: How to keep your secrets in a post-quantum world\nby Krist
in Lauter (Microsoft Research Redmond Labs) as part of Number Theory Web S
eminar\n\n\nAbstract\nAs we move towards a world which includes quantum co
mputers which exist at scale\, we are forced to consider the question of w
hat hard problems in mathematics our next generation of cryptographic syst
ems will be based on. Supersingular Isogeny Graphs were proposed for use
in cryptography in 2006 by Charles\, Goren\, and Lauter. Supersingular Is
ogeny Graphs are examples of Ramanujan graphs\, which are optimal expander
graphs. These graphs have the property that relatively short walks on t
he graph approximate the uniform distribution\, and for this reason\, walk
s on expander graphs are often used as a good source of randomness in comp
uter science. But the reason these graphs are important for cryptography
is that finding paths in these graphs\, i.e. routing\, is hard: there are
no known subexponential algorithms to solve this problem\, either classica
lly or on a quantum computer. For this reason\, cryptosystems based on th
e hardness of problems on Supersingular Isogeny Graphs are currently under
consideration for standardization in the NIST Post-Quantum Cryptography (
PQC) Competition. This talk will introduce these graphs\, the cryptograph
ic applications\, and the various algorithmic approaches which have been t
ried to attack these systems.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zeev Rudnick (Tel-Aviv University)
DTSTART:20200521T150000Z
DTEND:20200521T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/10
DESCRIPTION:Title: Prime lattice points in ovals\nby Zeev Rudnick (Tel-Aviv
University) as part of Number Theory Web Seminar\n\n\nAbstract\nThe study
of the number of lattice points in dilated regions has a long history\, wi
th several outstanding open problems. In this lecture\, I will describe a
new variant of the problem\, in which we study the distribution of lattice
points with prime coordinates. We count lattice points in which both coor
dinates are prime\, suitably weighted\, which lie in the dilate of a conve
x planar domain having smooth boundary\, with nowhere vanishing curvature.
We obtain an asymptotic formula\, with the main term being the area of th
e dilated domain\, and our goal is to study the remainder term. Assuming t
he Riemann Hypothesis\, we give a sharp upper bound\, and further assuming
that the positive imaginary parts of the zeros of the Riemann zeta functi
ons are linearly independent over the rationals allows us to give a formul
a for the value distribution function of the properly normalized remainder
term. (joint work with Bingrong Huang).\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Trevor Wooley (Purdue University)
DTSTART:20200528T150000Z
DTEND:20200528T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/11
DESCRIPTION:Title: Bracket quadratics\, Hua’s Lemma and Vinogradov’s mean va
lue theorem\nby Trevor Wooley (Purdue University) as part of Number Th
eory Web Seminar\n\n\nAbstract\nA little over a decade ago\, Ben Green pos
ed the problem of showing that all large integers are the sum of at most a
bounded number of bracket quadratic polynomials of the shape $n[n\\theta]
$\, for natural numbers $n$\, in which $\\theta$ is an irrational number s
uch as the square-root of 2. This was resolved in the PhD thesis of Vicky
Neale\, although no explicit bound was given concerning the number of vari
ables required to achieve success. In this talk we describe a version of H
ua’s lemma for this problem that can be applied via the Hardy-Littlewood
method to obtain a conclusion with 5 variables. The associated argument d
iffers according to whether $\\theta$ is a quadratic irrational or not. We
also explain how related versions of Hua’s lemma may be interpreted in
terms of discrete restriction variants of Vinogradov’s mean value theore
m\, thus providing a route to generalisation.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elon Lindenstrauss (Hebrew University of Jerusalem)
DTSTART:20200618T150000Z
DTEND:20200618T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/12
DESCRIPTION:Title: Diagonalizable flows\, joinings\, and arithmetic applications
\nby Elon Lindenstrauss (Hebrew University of Jerusalem) as part of Nu
mber Theory Web Seminar\n\n\nAbstract\nRigidity properties of higher rank
diagonalizable actions have proved to be powerful tools in understanding t
he distribution properties of rational tori in arithmetic quotients. Perha
ps the simplest\, and best known\, example of such an equidistribution que
stion is the equidistribution of CM points of a given discriminant on the
modular curve. The equidistribution of CM points was established by Duke u
sing analytic methods\, but for finer questions (and questions regarding e
quidistribution on higher rank spaces) the ergodic theoretic approach has
proved to be quite powerful.\n\nI will survey some of the results in this
direction\, including several results about joint distributions of collect
ions of points in product spaces by Aka\, Einsiedler\, Khayutin\, Shapira\
, Wieser and other researchers.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Maynard (University of Oxford)
DTSTART:20200702T150000Z
DTEND:20200702T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/13
DESCRIPTION:Title: Primes in arithmetic progressions to large moduli\nby Jam
es Maynard (University of Oxford) as part of Number Theory Web Seminar\n\n
\nAbstract\nHow many primes are there which are less than $x$ and congruen
t to $a$ modulo $q$? This is one of the most important questions in analyt
ic number theory\, but also one of the hardest - our current knowledge is
limited\, and any direct improvements require solving exceptionally diffic
ult questions to do with exceptional zeros and the Generalized Riemann Hyp
othesis!\n\nIf we ask for 'averaged' results then we can do better\, and p
owerful work of Bombieri and Vinogradov gives good answers for $q$ less th
an the square-root of $x$. For many applications this is as good as the Ge
neralized Riemann Hypothesis itself! Going beyond this 'square-root' barri
er is a notorious problem which has been achieved only in special situatio
ns\, perhaps most notably this was the key component in the work of Zhang
on bounded gaps between primes. I'll talk about recent work going beyond t
his barrier in some new situations. This relies on fun connections between
algebraic geometry\, spectral theory of automorphic forms\, Fourier analy
sis and classical prime number theory. The talk is intended for a general
audience.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Shparlinski (UNSW Sydney)
DTSTART:20200623T080000Z
DTEND:20200623T090000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/14
DESCRIPTION:Title: Weyl sums: large\, small and typical\nby Igor Shparlinski
(UNSW Sydney) as part of Number Theory Web Seminar\n\n\nAbstract\nAbstrac
t: While Vinogradov’s Mean Value Theorem\, in the form given by J. Bourg
ain\, C. Demeter and L. Guth (2016) and T. Wooley (2016-2019)\, gives an e
ssentially optimal result on the power moments of the Weyl sums \n$$\nS(u
\;N) =\\sum_{1\\le n \\le N} \\exp(2 \\pi i (u_1n+…+u_dn^d))\n$$\nwhere
$u = (u_1\,...\,u_d) \\in [0\,1)^d$\, very little is known about the dist
ribution\, or even existence\, of $u \\in [0\,1)^d$\, for which these sums
are very large\, or small\, or close to their average value $N^{1/2}$. In
this talk\, we describe recent progress towards these and some related qu
estions.\n\nWe also present some new bounds on $S(u\;N)$ which hold for al
most all $(u_i)_{i\\in I}$ and all $(u_j)_{j\\in J}$\, where $I \\cup J$ i
s a partition of $\\{1\,…\,\,d\\}$. These bounds improve similar results
of T. Wooley (2015). Our method also applies to binomial sums \n$$\nT(x\,
y\; N) = \\sum_{1\\le n \\le N} \\exp(2 \\pi i (xn+yn^d))\n$$\nwith $x\,y
\\in [0\,1)$\, in which case we improve some results of M.B. Erdogan and G
. Shakan (2019).\n\nThis is a joint work with Changhao Chen and Bryce Kerr
.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jennifer Balakrishnan (Boston University)
DTSTART:20200716T150000Z
DTEND:20200716T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/15
DESCRIPTION:Title: A tale of three curves\nby Jennifer Balakrishnan (Boston
University) as part of Number Theory Web Seminar\n\n\nAbstract\nWe will de
scribe variants of the Chabauty-Coleman method\nand quadratic Chabauty to
determine rational points on curves. In so\ndoing\, we will highlight some
recent examples where the techniques\nhave been used: this includes a pro
blem of Diophantus originally\nsolved by Wetherell and the problem of the
"cursed curve"\, the split\nCartan modular curve of level 13. This is join
t work with Netan Dogra\,\nSteffen Mueller\, Jan Tuitman\, and Jan Vonk.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuri Bilu (University of Bordeaux)
DTSTART:20200611T150000Z
DTEND:20200611T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/16
DESCRIPTION:Title: Trinomials\, singular moduli and Riffaut's conjecture\nby
Yuri Bilu (University of Bordeaux) as part of Number Theory Web Seminar\n
\n\nAbstract\nRiffaut (2019) conjectured that a singular modulus of degree
h>2 cannot be a root of a trinomial with rational coefficients. We show t
hat this conjecture follows from the GRH\, and obtain partial unconditiona
l results. A joint work with Florian Luca and Amalia Pizarro.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lillian Pierce (Duke University)
DTSTART:20200709T150000Z
DTEND:20200709T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/17
DESCRIPTION:Title: On Bourgain’s counterexample for the Schrödinger maximal f
unction\nby Lillian Pierce (Duke University) as part of Number Theory
Web Seminar\n\n\nAbstract\nThere is a long and visible history of applicat
ions of analytic methods to number theory. More recently we are starting t
o recognize applications of number-theoretic methods to analysis. In this
talk we will describe an important recent application in this direction. \
n\nIn 1980\, Carleson asked a question in PDE's: for what class of initial
data functions does a pointwise a.e. convergence result hold for the solu
tion of the linear Schrödinger equation? Over the next decades\, many peo
ple developed counterexamples to show “necessary conditions\,” and on
the other hand positive results to show “sufficient conditions.” In 20
16 Bourgain wrote a 3-page paper using facts from number theory to constru
ct a family of counterexamples. A 2019 Annals paper of Du and Zhang then r
esolved the question by proving positive results that push the “sufficie
nt conditions” to meet Bourgain’s “necessary conditions."\n\nBourgai
n’s construction was regarded as somewhat mysterious. In this talk\, we
give an overview of how to rigorously derive Bourgain’s construction usi
ng ideas from number theory. Our strategy is to start from “zero knowled
ge" and gradually optimize the set-up to arrive at the final counterexampl
e. This talk will be broadly accessible.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bas Edixhoven (Leiden University)
DTSTART:20200526T080000Z
DTEND:20200526T090000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/18
DESCRIPTION:Title: Geometric quadratic Chabauty\nby Bas Edixhoven (Leiden Un
iversity) as part of Number Theory Web Seminar\n\n\nAbstract\nJoint work w
ith Guido Lido (see arxiv preprint). Determining all rational points on a
curve of genus at least $2$ can be difficult. Chabauty's method (1941) is
to intersect\, for a prime number p\, in the p-adic Lie group of $p$-adic
points of the jacobian\, the closure of the Mordell-Weil group with the p-
adic points of the curve. If the Mordell-Weil rank is less than the genus
then this method has never failed. Minhyong Kim's non-abelian Chabauty pro
gramme aims to remove the condition on the rank. The simplest case\, calle
d quadratic Chabauty\, was developed by Balakrishnan\, Dogra\, Mueller\, T
uitman and Vonk\, and applied in a tour de force to the so-called cursed c
urve (rank and genus both $3$). Our work gives a version of this method th
at uses only `simple algebraic geometry' (line bundles over the jacobian a
nd models over the integers). For the talk\, no knowledge of all this alge
braic geometry is required\, it will be accessible to all number theorists
.\n\nReferences: https://arxiv.org/abs/1910.10752\nArizona Winter School 2
020: http://swc.math.arizona.edu/index.html\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph H. Silverman (Brown University)
DTSTART:20200730T150000Z
DTEND:20200730T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/19
DESCRIPTION:Title: More Tips on Keeping Secrets in a Post-Quantum World: Lattice
-Based Cryptography\nby Joseph H. Silverman (Brown University) as part
of Number Theory Web Seminar\n\n\nAbstract\nWhat do internet commerce\, o
nline banking\, and updates to your phone apps have in common? All of the
m depend on modern public key cryptography for security. For example\, th
ere is the RSA cryptosystem that is used by many internet browsers\, and
there is the elliptic curve based ECDSA digital signature scheme that is
used in many applications\, including Bitcoin. All of these cryptographic
construction are doomed if/when someone (NSA? Russia? China?) builds
a full-scale operational quantum computer. It hasn't happened yet\, as fa
r as we know\, but there are vast resources being thrown at the problem\,
and slow-but-steady progress is being made. So the search is on for cryp
tographic algorithms that are secure against quantum computers. The fir
st part of my talk will be a mix of math and history and prognostication
centered around the themes of quantum computers and public key cryptograp
hy. The second part will discuss cryptographic constructions based on har
d lattice problems\, which is one of the approaches being proposed to bui
ld a post-quantum cryptographic infrastructure.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kaisa Matomäki (University of Turku)
DTSTART:20200602T080000Z
DTEND:20200602T090000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/20
DESCRIPTION:Title: Multiplicative functions in short intervals revisited\nby
Kaisa Matomäki (University of Turku) as part of Number Theory Web Semina
r\n\n\nAbstract\nA few years ago Maksym Radziwill and I showed that the av
erage of a multiplicative function in almost all very short intervals $[x\
, x+h]$ is close to its average on a long interval $[x\, 2x]$. This result
has since been utilized in many applications.\n\nIn a work in progress th
at I will talk about\, Radziwill and I revisit the problem and generalise
our result to functions which vanish often as well as prove a power-saving
upper bound for the number of exceptional intervals (i.e. we show that th
ere are $O(X/h^\\kappa)$ exceptional $x \\in [X\, 2X]$). \n\nWe apply this
result for instance to studying gaps between norm forms of an arbitrary n
umber field.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bjorn Poonen (MIT)
DTSTART:20200806T150000Z
DTEND:20200806T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/21
DESCRIPTION:Title: Tetrahedra with rational dihedral angles\nby Bjorn Poonen
(MIT) as part of Number Theory Web Seminar\n\n\nAbstract\nIn 1895\, Hill
discovered a 1-parameter family of tetrahedra whose dihedral angles are al
l rational multiples of $\\pi$. In 1976\, Conway and Jones related the pro
blem of finding all such tetrahedra to solving a polynomial equation in ro
ots of unity. Many previous authors have solved polynomial equations in ro
ots of unity\, but never with more than $12$ monomials\, and the Conway-Jo
nes polynomial has $105$ monomials! I will explain the method we use to so
lve it and our discovery that the full classification consists of two $1$-
parameter families and an explicit finite list of sporadic tetrahedra.\n\n
Building on this work\, we classify all configurations of vectors in $\\R^
3$ such that the angle between each pair is a rational multiple of $\\pi$.
Sample result: Ignoring trivial families and scalar multiples\, any confi
guration with more than $9$ vectors is contained in a particular $15$-vect
or configuration. \n\nThis is joint work with Kiran Kedlaya\, Alexander K
olpakov\, and Michael Rubinstein.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Harald Andrés Helfgott (Göttingen/CNRS (IMJ))
DTSTART:20200616T080000Z
DTEND:20200616T090000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/22
DESCRIPTION:Title: Optimality of the logarithmic upper-bound sieve\, with explic
it estimates\nby Harald Andrés Helfgott (Göttingen/CNRS (IMJ)) as pa
rt of Number Theory Web Seminar\n\n\nAbstract\nAt the simplest level\, an
upper bound sieve of Selberg type is a choice of $\\rho(d)$\, $d\\le D$\,
with $\\rho(1)=1$\, such that\n$$\nS = \\sum_{n\\leq N} \\left(\\sum_{d|n}
\\mu(d) \\rho(d)\\right)^2\n$$\nis as small as possible.\n\nThe optimal c
hoice of $\\rho(d)$ for given $D$ was found by Selberg. However\, for seve
ral applications\, it is better to work with functions $\\rho(d)$ that are
scalings of a given continuous or monotonic function $\\eta$. The questio
n is then what is the best function $\\eta$\, and how does $S$ for given $
\\eta$ and $D$ compares to $S$ for Selberg's choice.\n\nThe most common ch
oice of eta is that of Barban-Vehov (1968)\, which gives an $S$ with the s
ame main term as Selberg's $S$. We show that Barban and Vehov's choice is
optimal among all $\\eta$\, not just (as we knew) when it comes to the mai
n term\, but even when it comes to the second-order term\, which is negati
ve and which we determine explicitly.\n\nThis is joint work with Emanuel C
arneiro\, Andrés Chirre and Julian Mejía-Cordero.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christopher Skinner (Princeton University)
DTSTART:20200820T150000Z
DTEND:20200820T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/23
DESCRIPTION:Title: Solving diagonal diophantine equations over general $p$-adic
fields\nby Christopher Skinner (Princeton University) as part of Numbe
r Theory Web Seminar\n\n\nAbstract\nThis talk will explain a proof that a
system of $r$ diagonal equations\n$$\na_{i\,1}x_1^d + \\cdots +a_{i\,s} x_
s^d = 0 \,\\quad i = 1\,...\,r\n$$\nwith coefficients in a $p$-adic field
$K$ has a non-trivial solution in $K$ if the number of variables $s$ excee
ds $3r^2d^2$ (if $p > 2$) or $8r^2d^2$ (if $p=2$). This is the first boun
d that holds uniformly for all $p$-adic fields K and that is polynomial in
$r$ or $d$. The methods -- and talk -- are elementary.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hector Pasten (Pontificia Universidad Católica de Chile)
DTSTART:20200827T150000Z
DTEND:20200827T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/24
DESCRIPTION:Title: A Chabauty-Coleman bound for hyperbolic surfaces in abelian t
hreefolds\nby Hector Pasten (Pontificia Universidad Católica de Chile
) as part of Number Theory Web Seminar\n\n\nAbstract\nA celebrated result
of Coleman gives a completely explicit version of Chabauty's finiteness th
eorem for rational points in hyperbolic curves over a number field\, by a
study of zeros of p-adic analytic functions. After several developments ar
ound this result\, the problem of proving an analogous explicit bound for
higher dimensional subvarieties of abelian varieties remains elusive. In t
his talk I'll sketch the proof of such a bound for hyperbolic surfaces con
tained in abelian threefolds. This is joint work with Jerson Caro.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Özlem Imamoglu (ETH Zürich)
DTSTART:20200917T150000Z
DTEND:20200917T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/25
DESCRIPTION:Title: A class number formula of Hurwitz\nby Özlem Imamoglu (ET
H Zürich) as part of Number Theory Web Seminar\n\n\nAbstract\nIn a little
known paper Hurwitz gave an infinite series representation for the cla
ss number of positive definite binary quadratic forms In this talk I will
report on joint work with W. Duke and A. Toth where we show how the id
eas of Hurwitz can be applied in other settings\, in particular to give a
formula for the class number of binary cubic forms.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emmanuel Breuillard (University of Cambridge)
DTSTART:20200924T150000Z
DTEND:20200924T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/26
DESCRIPTION:Title: A subspace theorem for manifolds\nby Emmanuel Breuillard
(University of Cambridge) as part of Number Theory Web Seminar\n\n\nAbstra
ct\nIn the late 90's Kleinbock and Margulis solved a long-standing conject
ure due to Sprindzuk regarding diophantine approximation on submanifolds o
f $\\R^n$. Their method used homogeneous dynamics via the so-called non-di
vergence estimates for unipotent flows on the space of lattices. In this t
alk I will explain how these ideas\, combined with a certain understanding
of the geometry at the heart of Schmidt's subspace theorem\, in particula
r the notion of Harder-Narasimhan filtration\, leads to a metric version o
f the subspace theorem\, where the linear forms are allowed to depend on a
parameter. This subspace theorem for manifolds allows to quickly compute
certain diophantine exponents\, and it leads to several generalizations of
the Kleinbock-Margulis results in a variety of contexts. Joint work with
Nicolas de Saxcé.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bianca Viray (University of Washington)
DTSTART:20200910T150000Z
DTEND:20200910T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/27
DESCRIPTION:Title: Existence of quadratic points on intersections of quadrics\nby Bianca Viray (University of Washington) as part of Number Theory Web
Seminar\n\n\nAbstract\nSpringer's theorem and the Amer-Brumer theorem tog
ether imply that intersections of two quadrics have a rational point if an
d only if they have a $0$-cycle of degree $1$. In this talk\, we conside
r whether this statement can be strengthened in the case when there is no
rational point\, namely whether 1) the least degree of a $0$-cycle can be
bounded\, and 2) whether there is an effective $0$-cycle of this degree.
We report on results in this direction\, paying particular attention to t
he case of local and global fields. This is joint work with Brendan Creu
tz.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carl Pomerance (Dartmouth College)
DTSTART:20200813T150000Z
DTEND:20200813T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/28
DESCRIPTION:Title: Practical numbers\nby Carl Pomerance (Dartmouth College)
as part of Number Theory Web Seminar\n\n\nAbstract\nA practical number $n$
is one where each number up to $n$ can be expressed as a subset sum of $n
$'s positive divisors. It seems that Fibonacci was interested in them sinc
e they have the property that all fractions $m/n$ with $m < n$ can be writ
ten as a sum of distinct unit fractions with denominators dividing $n$. W
ith similar considerations in mind\, Srinivasan in 1948 coined the term "p
ractical". There has been quite a lot of effort to study their distributio
n\, effort which has gone hand in hand with the development of the anatomy
of integers. After work of Tenenbaum\, Saias\, and Weingartner\, we now
know the "Practical Number Theorem": the number of practical numbers up to
$x$ is asymptotically $cx/log x$\, where $c= 1.33607...$. In this talk I
'll discuss some recent developments\, including work of Thompson who cons
idered the allied concept of $\\phi$-practical numbers $n$ (the polynomial
$t^n-1$ has divisors over the integers of every degree up to $n$) and the
proof (joint with Weingartner) of a conjecture of Margenstern that each l
arge odd number can be expressed as a sum of a prime and a practical numbe
r.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:René Schoof (Università di Roma “Tor Vergata”)
DTSTART:20200707T080000Z
DTEND:20200707T090000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/29
DESCRIPTION:Title: Abelian varieties over ${\\bf Q}(\\sqrt{97})$ with good reduc
tion everywhere\nby René Schoof (Università di Roma “Tor Vergata
”) as part of Number Theory Web Seminar\n\n\nAbstract\nUnder assumption
of the Generalized Riemann Hypothesis we show that every abelian variety o
ver ${\\bf Q}(\\sqrt{97})$ with good reduction everywhere is isogenous to
a power of a certain $3$-dimensional modular abelian variety.\n\n(joint wi
th Lassina Dembele)\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kannan Soundararajan (Stanford University)
DTSTART:20200630T000000Z
DTEND:20200630T010000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/30
DESCRIPTION:Title: Equidistribution from the Chinese Remainder Theorem\nby K
annan Soundararajan (Stanford University) as part of Number Theory Web Sem
inar\n\n\nAbstract\nSuppose for each prime $p$ we are given a set $A_p$ (p
ossibly empty) of residue classes mod $p$. Use these and the Chinese Rema
inder Theorem to form a set $A_q$ of residue classes mod $q$\, for any int
eger $q$. Under very mild hypotheses\, we show that for a typical integer
$q$\, the residue classes in $A_q$ will become equidistributed. The prot
otypical example (which this generalises) is Hooley's theorem that the roo
ts of a polynomial congruence mod $n$ are equidistributed on average over
$n$. I will also discuss generalisations of such results to higher dimens
ions\, and when restricted to integers with a given number of prime factor
s. (Joint work with Emmanuel Kowalski.)\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jordan Ellenberg (University of Wisconsin–Madison)
DTSTART:20200723T150000Z
DTEND:20200723T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/31
DESCRIPTION:Title: What’s up in arithmetic statistics\nby Jordan Ellenberg
(University of Wisconsin–Madison) as part of Number Theory Web Seminar\
n\n\nAbstract\nIf not for a global pandemic\, a bunch of mathematicians wo
uld have gathered in Germany to talk about what’s going on in the geomet
ry of arithmetic statistics\, which I would roughly describe as “methods
from arithmetic geometry brought to bear on probabilistic questions about
arithmetic objects". What does the maximal unramified extension of a rand
om number field look like? What is the probability that a random elliptic
curve has a $2$-Selmer group of rank 100? How do you count points on a st
ack? I’ll give a survey of what’s happening in questions in this area\
, trying to emphasize open questions.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ken Ono (University of Virginia)
DTSTART:20200714T000000Z
DTEND:20200714T010000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/32
DESCRIPTION:Title: Variants of Lehmer's speculation for newforms\nby Ken Ono
(University of Virginia) as part of Number Theory Web Seminar\n\n\nAbstra
ct\nIn the spirit of Lehmer's unresolved speculation on the nonvanishing o
f Ramanujan's tau-function\, it is natural to ask whether a fixed integer
is a value of τ(n)\, or is a Fourier coefficient of any given newform. I
n joint work with J. Balakrishnan\, W. Craig\, and W.-L. Tsai\, the speake
r has obtained some results that will be described here. For example\, inf
initely many spaces are presented for which the primes ℓ≤37 are not ab
solute values of coefficients of any newforms with integer coefficients. F
or Ramanujan’s tau-function\, such results imply\, for n>1\, that\n\nτ(
n)∉{±1\,±3\,±5\,±7\,±13\,±17\,−19\,±23\,±37\,±691}.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wadim Zudilin (Radboud University Nijmegen)
DTSTART:20200721T080000Z
DTEND:20200721T090000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/33
DESCRIPTION:Title: Irrationality through an irrational time\nby Wadim Zudili
n (Radboud University Nijmegen) as part of Number Theory Web Seminar\n\n\n
Abstract\nAfter reviewing some recent development and achievements related
to diophantine problems of the values of Riemann's zeta function and gene
ralized polylogarithms (not all coming from myself!)\, I will move the foc
us to $\\pi=3.1415926\\dots$ and its rational approximations. Specifically
\, I will discuss a construction of rational approximations to the number
that leads to the record irrationality measure of $\\pi$. The talk is base
d on joint work with Doron Zeilberger.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin Ford (University of Illinois at Urbana-Champaign)
DTSTART:20200903T150000Z
DTEND:20200903T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/34
DESCRIPTION:Title: Prime gaps\, probabilistic models\, the interval sieve\, Hard
y-Littlewood conjectures and Siegel zeros\nby Kevin Ford (University o
f Illinois at Urbana-Champaign) as part of Number Theory Web Seminar\n\n\n
Abstract\nMotivated by a new probabilistic interpretation of the Hardy-Lit
tlewood $k$-tuples conjectures\, we introduce a new probabilistic model of
the primes and make a new conjecture about the largest gaps between the p
rimes below $x$. Our bound depends on a property of the interval sieve whi
ch is not well understood. We also show that any sequence of integers whic
h satisfies a sufficiently uniform version of the Hardy-Littlewood conject
ures must have large gaps of a specific size. Finally\, assuming that Sieg
el zeros exist we show the existence of gaps between primes which are subs
tantially larger than the gaps which are known unconditionally. Much of th
is work is joint with Bill Banks and Terry Tao.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wei Ho (University of Michigan)
DTSTART:20201001T150000Z
DTEND:20201001T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/35
DESCRIPTION:Title: The Hasse local-to-global principle for some genus one curves
\nby Wei Ho (University of Michigan) as part of Number Theory Web Semi
nar\n\n\nAbstract\nThe Hasse principle is a useful guiding philosophy in a
rithmetic geometry that relates "global" questions to analogous "local" qu
estions\, which are often easier to understand. A simple incarnation of th
e Hasse principle says that a given polynomial equation has a solution in
the rational numbers (i.e.\, is "globally soluble") if and only if it has
a solution in the real numbers and in the p-adic numbers for all primes p
(i.e.\, is "everywhere locally soluble"). While this principle holds for m
any "simple" such polynomials\, it is a very difficult question to classif
y the polynomials (or more generally\, algebraic varieties) for which the
principle holds or fails.\n\nIn this talk\, we will discuss problems relat
ed to the Hasse principle for some classes of varieties\, with a special f
ocus on genus one curves given by bihomogeneous polynomials of bidegree $(
2\,2)$ in $\\mathbb{P}^1 \\times \\mathbb{P}^1$. For example\, we will des
cribe how to compute the proportion of these curves that are everywhere lo
cally soluble (joint work with Tom Fisher and Jennifer Park)\, and we will
explain why the Hasse principle fails for a positive proportion of these
curves\, by comparing the average sizes of $2$- and $3$-Selmer groups for
a family of elliptic curves with a marked point (joint work with Manjul Bh
argava).\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Philippe Michel (EPFL)
DTSTART:20201008T150000Z
DTEND:20201008T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/36
DESCRIPTION:Title: Simultaneous reductions of CM elliptic curves\nby Philipp
e Michel (EPFL) as part of Number Theory Web Seminar\n\n\nAbstract\nLet $E
$ be an elliptic curve with CM by the imaginary quadratic order $O_D$ of d
iscriminant $D<0$. Given $p$ a prime \; if $p$ is inert or ramified in the
quadratic field generated by $\\sqrt D$ then $E$ has supersingular reduct
ion at a(ny) fixed place above $p$. By a variant of Duke’s equidistribut
ion theorem\, as $D$ grows along such discriminants\, the proportion of CM
elliptic curves with CM by $O_D$ whose reduction at such place is a given
supersingular curve converge to a natural (non-zero) limit. A further ste
p is to fix several (distinct) primes $p_1\,\\cdots\,p_s$ and to look for
the proportion of CM curves whose reduction above each of these primes is
prescribed. In this talk\, we will explain how a powerful result of Einsie
dler and Lindenstrauss classifying joinings of rank $2$ actions on product
s of locally homogeneous spaces implies that as $D$ grows along adequate s
ubsequences of negative discriminants\, this proportion converge to the pr
oduct of the limits for each individual $p_i$ (a sort of asymptotic Chines
e Reminder Theorem for reductions of CM elliptic curves if you wish). This
is joint work with M. Aka\, M. Luethi and A.Wieser. If time permits\, we
will also describe a further refinement -- obtained with the additional co
llaboration of R. Menares — of these equidistribution results for the fo
rmal groups attached to these curves.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Umberto Zannier (Scuola Normale Superiore Pisa)
DTSTART:20200901T090000Z
DTEND:20200901T100000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/37
DESCRIPTION:Title: Torsion in elliptic familes and applications to billiards
\nby Umberto Zannier (Scuola Normale Superiore Pisa) as part of Number The
ory Web Seminar\n\n\nAbstract\nWe shall consider elliptic pencils\, of whi
ch the best-known example is probably the Legendre family $L_t$: $y^2=x(x-
1)(x-t)$ where $t$ is a parameter. Given a section $P(t)$ (i.e. a family o
f points on $L_t$ depending on $t$) it is an issue to study the set of c
omplex $b$ such that $P(b)$ is torsion on $L_b$. We shall recall a number
of results on the nature of this set. Then we shall present some applicati
ons (obtained jointly with P. Corvaja) to elliptical billiards. For instan
ce\, if two players hit the same ball with directions forming a given angl
e in $(0\,\\pi)$\, there are only finitely many cases for which both billi
ard trajectories are periodic.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cameron L. Stewart (University of Waterloo)
DTSTART:20201015T150000Z
DTEND:20201015T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/38
DESCRIPTION:Title: On integers represented by binary forms\nby Cameron L. St
ewart (University of Waterloo) as part of Number Theory Web Seminar\n\n\nA
bstract\nWe shall discuss the following results which are joint work with
Stanley Xiao.\n\nLet $F(x\,y)$ be a binary form with integer coefficients\
, degree $d(>2)$ and non-zero discriminant. There is a positive number $C(
F)$ such that the number of integers of absolute value at most $Z$ which a
re represented by $F$ is asymptotic to $C(F)Z^{2/d}$.\n\nLet $k$ be an int
eger with $k>1$ and suppose that there is no prime $p$ such that $p^k$ div
ides $F(a\,b)$ for all pairs of integers $(a\,b)$. Then\, provided that $k
$ exceeds $7d/18$ or $(k\,d)$ is $(2\,6)$ or $(3\,8)$\, there is a positiv
e number $C(F\,k)$ such that the number of $k$-free integers of absolute v
alue at most $Z$ which are represented by $F$ is asymptotic to $C(F\,k)Z^{
2/d}$.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergei Konyagin (Steklov Institute of Mathematics)
DTSTART:20201022T150000Z
DTEND:20201022T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/39
DESCRIPTION:Title: A construction of A. Schinzel - many numbers in a short inter
val without small prime factors\nby Sergei Konyagin (Steklov Institute
of Mathematics) as part of Number Theory Web Seminar\n\n\nAbstract\nHardy
and Littlewood (1923) conjectured that for any integers $x\,y\\ge2$\n$$\n
\\pi(x+y) \\le \\pi(x) + \\pi(y). \\qquad\\qquad\\qquad (1)\n$$\n\nLet us
call a set $\\{b_1\,\\dots\,b_k\\}$ of integers admissible if for each\npr
ime $p$ there is some congruence class $\\bmod p$ which contains none\nof
the integers $b_i$. The prime $k$-tuple conjecture states that if a set \n
$\\{b_1\,\\dots\,b_k\\}$ is admissible\, then there exist infinitely many
\nintegers $n$ for which all the numbers $n+b_1\,\\dots\,n+b_k$ are primes
.\n\nLet $x$ be a positive integer and $\\rho^*(x)$ be the maximum number\
nof integers in any interval $(y\,y+x]$ (with no restriction on $y$)\nwhic
h are relatively prime to all positive integers $\\le x$.\nThe prime $k$-t
uple conjecture implies that\n$$\\max_{y\\ge x}(\\pi(x+y)-\\pi(y))=\\limsu
p_{y\\ge x} (\\pi(x+y)-\\pi(y))=\\rho^*(x).$$\n\nHensley and Richards (197
4) proved that\n$$\\rho^*(x) - \\pi(x) \\ge(\\log 2- o(1)) x(\\log x)^{-2}
\\quad(x\\to\\infty).$$\nTherefore\, (1) is not compatible with the prime
$k$-tuple\nconjecture. Using a construction of Schinzel we show that\n$$\\
rho^*(x) - \\pi(x) \\ge((1/2)- o(1)) x(\\log x)^{-2}\\log\\log\\log x\\qua
d(x\\to\\infty).$$\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dzmitry Badziahin (University of Sydney)
DTSTART:20200915T090000Z
DTEND:20200915T100000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/40
DESCRIPTION:Title: Approximation by algebraic numbers\nby Dzmitry Badziahin
(University of Sydney) as part of Number Theory Web Seminar\n\n\nAbstract\
nIn this talk we discuss the approximation of transcendental numbers by al
gebraic numbers of given degree and bounded height. More precisely\, for a
ny real number $x$\, by $w_n^*(x)$ we define the supremum of all positive
real values $w$ such that the inequality\n $$ |x - a| < H(a)^{-w-1}$
$\nhas infinitely many solutions in algebraic real numbers $a$ of degree a
t most $n$. Here $H(a)$ means the naive height of the minimal polynomial i
n $\\Z[x]$ with coprime coefficients. In 1961\, Wirsing asked: is it true
that the quantity $w_n^*(x)$ is at least n for all transcendental $x$? Apa
rt from partial results for small values of $n$\, this problem still remai
ns open. Wirsing himself managed to establish the lower bound of the form
$w_n^*(x) \\ge n/2+1 - o(1)$. Until recently\, the only improvements to th
is bound were in terms of $O(1)$. I will talk about our resent work with S
chleischitz where we managed to improve the bound by a quantity of the siz
e $O(n)$. More precisely\, we show that $w_n^*(x) > n/\\sqrt{3}$.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Will Sawin (Columbia University)
DTSTART:20201029T160000Z
DTEND:20201029T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/41
DESCRIPTION:Title: The distribution of prime polynomials over finite fields\
nby Will Sawin (Columbia University) as part of Number Theory Web Seminar\
n\n\nAbstract\nMany conjectures in number theory have analogues for polyno
mials in one variable over a finite field. In recent works with Mark Shust
erman\, we proved analogues of two conjectures about prime numbers - the t
win primes conjecture and the conjecture that there are infinitely many pr
imes of the form $n^2+1$. I will describe these results and explain some o
f the key ideas in the proofs\, which combine classical analytic methods\,
elementary algebraic manipulations\, and geometric methods.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dragos Ghioca (University of British Columbia)
DTSTART:20201201T010000Z
DTEND:20201201T020000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/42
DESCRIPTION:Title: A couple of conjectures in arithmetic dynamics over fields of
positive characteristic\nby Dragos Ghioca (University of British Colu
mbia) as part of Number Theory Web Seminar\n\n\nAbstract\nThe Dynamical Mo
rdell-Lang Conjecture predicts the structure of the intersection between a
subvariety $V$ of a variety $X$ defined over a field $K$ of characteristi
c $0$ with the orbit of a point in $X(K)$ under an endomorphism $\\Phi$ of
$X$. The Zariski dense conjecture provides a dichotomy for any rational s
elf-map $\\Phi$ of a variety $X$ defined over an algebraically closed fiel
d $K$ of characteristic $0$: either there exists a point in $X(K)$ with a
well-defined Zariski dense orbit\, or $\\Phi$ leaves invariant some non-co
nstant rational function $f$. For each one of these two conjectures we for
mulate an analogue in characteristic $p$\; in both cases\, the presence of
the Frobenius endomorphism in the case $X$ is isotrivial creates signific
ant complications which we will explain in the case of algebraic tori.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ilya D. Shkredov (Steklov Mathematical Institute\, Moscow)
DTSTART:20200922T090000Z
DTEND:20200922T100000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/43
DESCRIPTION:Title: Zaremba's conjecture and growth in groups\nby Ilya D. Shk
redov (Steklov Mathematical Institute\, Moscow) as part of Number Theory W
eb Seminar\n\n\nAbstract\nZaremba's conjecture belongs to the area of cont
inued fractions. It predicts that for any given positive integer $q$ there
is a positive $a$\, $a < q$\, $(a\,q)=1$ such that all partial quotients
$b_j$ in its continued fractions expansion $a/q = 1/b_1+1/b_2 +...+ 1/b_s
$ are bounded by five. At the moment the question is widely open although
the area has a rich history of works by Korobov\, Hensley\, Niederreiter\,
Bourgain and many others. We survey certain results concerning this hypot
hesis and show how growth in groups helps to solve different relaxations o
f Zaremba's conjecture. In particular\, we show that a deeper hypothesis o
f Hensley concerning some Cantor-type set with the Hausdorff dimension $>1
/2$ takes place for the so-called modular form of Zaremba's conjecture.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Smith (Harvard University)
DTSTART:20201006T000000Z
DTEND:20201006T010000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/44
DESCRIPTION:Title: Selmer groups and a Cassels-Tate pairing for finite Galois mo
dules\nby Alexander Smith (Harvard University) as part of Number Theor
y Web Seminar\n\n\nAbstract\nI will discuss some new results on the struct
ure of Selmer groups of finite Galois modules over global fields. Tate's d
efinition of the Cassels-Tate pairing can be extended to a pairing on such
Selmer groups with little adjustment\, and many of the fundamental proper
ties of the Cassels-Tate pairing can be reproved with new methods in this
setting. I will also give a general definition of the theta/Mumford group
and relate it to the structure of the Cassels-Tate pairing\, generalizing
work of Poonen and Stoll.\n\nAs one application of this theory\, I will pr
ove an elementary result on the symmetry of the class group pairing for nu
mber fields with many roots of unity and connect this to the work of mine
and others on class group statistics.\n\nThis work is joint with Adam Morg
an.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julie Tzu-Yueh Wang (Academia Sinica)
DTSTART:20200929T000000Z
DTEND:20200929T010000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/45
DESCRIPTION:Title: Pisot's $d$-th root's conjecture for function fields and its
complex analog\nby Julie Tzu-Yueh Wang (Academia Sinica) as part of Nu
mber Theory Web Seminar\n\n\nAbstract\nPisot's $d$-th root's conjecture\,
proved by Zannier in 2000\, can be stated as follows.\nLet $b$ be a lin
ear recurrence \nover a number field $k$\, and $d\\ge2$ be an integer. Su
ppose that\n$b(n)$ is the $d$-th power of some element in $k$ for all but
finitely\nmany $n$. Then there exists a linear recurrence $a$\nover $\\ove
rline{k}$ such that $a(n)^{d}=b(n)$ for all $n$.\n\n\nIn this talk\, we p
ropose a function-field analog of this result and prove it under some ``n
on-triviality''\nassumption. We relate the problem to a result of Pasten
-Wang on B\\"uchi's $d$-th power problem and develop a function-field
GCD estimate for multivariable polynomials with ``small coefficients" eval
uating at $S$-units arguments. We will also discuss its complex analog in
the notion of (generalized Ritt's) exponential polynomials. \n\nThis
is a joint work with Ji Guo and Chia-Liang Sun.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maryna Viazovska (EPFL)
DTSTART:20200908T080000Z
DTEND:20200908T090000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/46
DESCRIPTION:Title: Universal optimality\, Fourier interpolation\, and modular in
tegrals\nby Maryna Viazovska (EPFL) as part of Number Theory Web Semin
ar\n\n\nAbstract\nIn this lecture we will show that the E8 and Leech latt
ices minimize energy for a wide class of potential functions. This theorem
implies recently proven optimality of E8 and Leech lattices as sphere pac
kings and broadly generalizes it to long-range interactions. The key ingre
dient of the proof is sharp linear programming bounds. Construction of the
optimal auxiliary functions attaining these bounds is based on a new inte
rpolation theorem. This is joint work with Henry Cohn\, Abhinav Kumar\, St
ephen D. Miller\, and Danylo Radchenko.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Par Kurlberg (KTH)
DTSTART:20201105T160000Z
DTEND:20201105T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/47
DESCRIPTION:Title: Distribution of lattice points on hyperbolic circles\nby
Par Kurlberg (KTH) as part of Number Theory Web Seminar\n\n\nAbstract\nWe
study the distribution of lattice points lying on expanding circles in the
hyperbolic plane. The angles of lattice points arising from the orbit of
the modular group $\\mathrm{PSL}(2\,\\Z)$\, and lying on hyperbolic circle
s centered at i\, are shown to be equidistributed for generic radii (among
the ones that contain points). We also show that angles fail to equidistr
ibute on a thin set of exceptional radii\, even in the presence of growing
multiplicity. Surprisingly\, the distribution of angles on hyperbolic cir
cles turns out to be related to the angular distribution of euclidean latt
ice points lying on circles in the plane\, along a thin subsequence of rad
ii. This is joint work with D. Chatzakos\, S. Lester and I. Wigman.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Gorodnik (University of Zurich)
DTSTART:20201013T090000Z
DTEND:20201013T100000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/49
DESCRIPTION:Title: Arithmetic approach to the spectral gap problem\nby Alexa
nder Gorodnik (University of Zurich) as part of Number Theory Web Seminar\
n\n\nAbstract\nThe spectral gap is an analytic property of group actions w
hich can be described as absence of "almost invariant vectors" or more qua
ntitatively in terms of norm bounds for suitable averaging operators. In t
he setting of homogeneous spaces this property also has a profound number-
theoretic meaning since it is closely related to understanding the automor
phic representations. In this talk we survey some previous results about t
he spectral gap property and describe new approaches to deriving upper and
lower bounds for the spectral gap.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chantal David (Concordia University)
DTSTART:20201119T160000Z
DTEND:20201119T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/50
DESCRIPTION:Title: CANCELLED--Moments and non-vanishing of cubic Dirichlet $L$-f
unctions at $s=\\frac{1}{2}$\nby Chantal David (Concordia University)
as part of Number Theory Web Seminar\n\n\nAbstract\nA famous conjecture of
Chowla predicts that $L(\\frac{1}{2}\,\\chi)\\ne 0$ for all Dirichlet $L$
-functions\nattached to primitive characters $\\chi$. It was conjectured f
irst in the case where $\\chi$ is a quadratic\ncharacter\, which is the mo
st studied case. For quadratic Dirichlet $L$-functions\, Soundararajan\npr
oved that at least 87.5% of the quadratic Dirichlet $L$-functions do not v
anish at $s=\\frac{1}{2}$.\nUnder GRH\, there are slightly stronger result
s by Ozlek and Snyder.\n\nWe present in this talk the first result showing
a positive proportion of cubic Dirichlet\n$L$-functions non-vanishing at
$s=\\frac{1}{2}$ for the non-Kummer case over function fields. This can\nb
e achieved by using the recent breakthrough work on sharp upper bounds for
moments of\nSoundararajan\, Harper and Lester-Radziwill. Our results woul
d transfer over number fields\,\nbut we would need to assume GRH in this c
ase.\n\nCANCELLED! There will be no talk this Thursday.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maksym Radziwill (California Institute of Technology)
DTSTART:20201210T160000Z
DTEND:20201210T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/51
DESCRIPTION:Title: The Fyodorov-Hiary-Keating conjecture\nby Maksym Radziwil
l (California Institute of Technology) as part of Number Theory Web Semina
r\n\n\nAbstract\nI will discuss recent progress on the Fyodorov-Hiary-Keat
ing conjecture on the distribution of the local maximum of the Riemann zet
a-function. This is joint work with Louis-Pierre Arguin and Paul Bourgade.
\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Masser (University of Basel)
DTSTART:20201112T160000Z
DTEND:20201112T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/52
DESCRIPTION:Title: Pencils of norm form equations and a conjecture of Thomas
\nby David Masser (University of Basel) as part of Number Theory Web Semin
ar\n\n\nAbstract\nWe consider certain one-parameter families of norm form
(and other) diophantine equations\, and we solve them completely and unifo
rmly for all sufficiently large positive integer values of the parameter (
everything effective)\, following a line started by Emery Thomas in 1990.
The new tool is a bounded height result from 2017 by Francesco Amoroso\, U
mberto Zannier and the speaker.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jörg Brüdern (University of Göttingen)
DTSTART:20201020T090000Z
DTEND:20201020T100000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/53
DESCRIPTION:Title: Harmonic analysis of arithmetic functions\nby Jörg Brüd
ern (University of Göttingen) as part of Number Theory Web Seminar\n\n\nA
bstract\nWe study arithmetic functions that are bounded in mean square\, a
nd simultaneously have a mean value over any arithmetic progression. A Bes
icovitch type norm makes the set of these functions a Banach space. We app
ly the Hardy-Littlewood (circle) method to analyse this space. This method
turns out to be a surprisingly flexible tool for this purpose. We obtain
several characterisations of limit periodic functions\, correlation formul
ae\, and we give some applications to Waring's problem and related topics.
Finally\, we direct the theory to the distribution of the arithmetic func
tions under review in arithmetic progressions\, with mean square results o
f Barban-Davenport-Halberstam type and related asymptotic formulae at the
focus of our attention. There is a rich literature on this last theme. Our
approach supersedes previous work in various ways\, and ultimately provid
es another characterisation of limit periodic functions: the variance over
arithmetic progression is atypically small if and only if the input funct
ion is limit periodic.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gal Binyamini (Weizmann Institute of Science)
DTSTART:20201027T100000Z
DTEND:20201027T110000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/54
DESCRIPTION:Title: Point counting for foliations in Diophantine geometry\nby
Gal Binyamini (Weizmann Institute of Science) as part of Number Theory We
b Seminar\n\n\nAbstract\nI will discuss "point counting" in two broad sens
es: counting the intersections between a trascendental variety and an alge
braic one\; and counting the number of algebraic points\, as a function of
degree and height\, on a transcendental variety. After reviewing the fund
amental results in this area - from the theory of o-minimal structures and
the Pila-Wilkie theorem\, I will restrict attention to the case that the
transcendental variety is given in terms of a leaf of an algebraic foliati
on\, and everything is defined over a number field. It turns out that in t
his case far stronger estimates can be obtained.\n\nApplying the above to
foliations associated to principal G-bundles on various moduli spaces\, ma
ny classical application of the Pila-Wilkie theorem can be sharpened and e
ffectivized. In particular I will discuss issues around effectivity and po
lynomial-time solvability for the Andre-Oort conjecture\, unlikely interse
ctions in abelian schemes\, and some related directions.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jacob Tsimerman (University of Toronto)
DTSTART:20201207T220000Z
DTEND:20201207T230000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/55
DESCRIPTION:Title: Bounding torsion in class group and families of local systems
\nby Jacob Tsimerman (University of Toronto) as part of Number Theory
Web Seminar\n\n\nAbstract\n(joint w/ Arul Shankar) We discuss a new method
to bound 5-torsion in class groups of quadratic fields using the refined
BSD conjecture for elliptic curves. The most natural “trivial” bound o
n the n-torsion is to bound it by the size of the entire class group\, for
which one has a global class number formula. We explain how to make sense
of the n-torsion of a class group intrinsically as a selmer group of a Ga
lois module. We may then similarly bound its size by the Tate-Shafarevich
group of an appropriate elliptic curve\, which we can bound using the BSD
conjecture. This fits into a general paradigm where one bounds selmer grou
ps of finite Galois modules by embedding into global objects\, and using c
lass number formulas. If time permits\, we explain how the function field
picture yields unconditional results and suggests further generalizations.
\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gisbert Wüstholz (ETH / University Zurich)
DTSTART:20201217T160000Z
DTEND:20201217T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/56
DESCRIPTION:Title: Baker's theory for $1$-motives\nby Gisbert Wüstholz (ETH
/ University Zurich) as part of Number Theory Web Seminar\n\n\nAbstract\n
From a historical point of view transcendence theory used to be a nice\nc
ollection of mostly particular results\, very difficult to find and to pro
ve. To find\nnumbers for which one has a chance to prove transcendence is
very difficult.\nTo state conjecture is not so difficult but in most cases
hopeless to prove.\nIn our lecture we try to draw a picture of quite far
reaching frames in the theory\nof motives which can put transcendence theo
ry into a more conceptual setting.\n\nLooking at periods of rational $1$-f
orms on varieties we realized that there is a\nmore conceptual background
behind the properties of these complex numbers \nthan had been thought so
far. The central question which I was trying for more than\nthree decades
to answer was to determine when a period is algebraic. A priori a period
is zero\, algebraic\nor transcendental\, no surprise! It is also not diffi
cult to give examples for cases when periods are algebraic.\nHowever the b
ig question was whether the examples are all examples. Quite recently\, pa
rtly jointly\nwith Annette Huber we developed a new transcendence theory w
ithin $1$-motives which extend commutative algebraic groups. One outcome w
as that algebraicity of periods has a very conceptual description\nand we
shall give a precise and surprisingly simple answer. \n\n Many questions
which were central in transcendence theory and with a long \nand famous h
istory turn out to get a general answer within the new theory. The class
ical work of Baker \nturns out to be a very special but seminal case.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jens Marklof (University of Bristol)
DTSTART:20201103T100000Z
DTEND:20201103T110000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/57
DESCRIPTION:Title: The three gap theorem in higher dimensions\nby Jens Markl
of (University of Bristol) as part of Number Theory Web Seminar\n\n\nAbstr
act\nTake a point on the unit circle and rotate it N times by a fixed angl
e. The N points thus generated partition the circle into N intervals. A be
autiful fact\, first conjectured by Hugo Steinhaus in the 1950s and proved
independently by Vera Sós\, János Surányi and Stanisław Świerczkowsk
i\, is that for any choice of N\, no matter how large\, these intervals ca
n have at most three distinct lengths. In this lecture I will explore an i
nterpretation of the three gap theorem in terms of the space of Euclidean
lattices\, which will produce various new results in higher dimensions\, i
ncluding gaps in the fractional parts of linear forms and nearest neighbou
r distances in multi-dimensional Kronecker sequences. The lecture is based
on joint work with Alan Haynes (Houston) and Andreas Strömbergsson (Upps
ala).\n\n1. Wikipedia\, https://en.wikipedia.org/wiki/Three-gap_theorem \n
\n2. J. Marklof and A. Strömbergsson\, The three gap theorem and the spac
e of lattices\, American Mathematical Monthly 124 (2017) 741-745 https://p
eople.maths.bris.ac.uk/~majm/bib/threegap.pdf\n\n3. A. Haynes and J. Markl
of\, Higher dimensional Steinhaus and Slater problems via homogeneous dyna
mics\, Annales scientifiques de l'Ecole normale superieure 53 (2020) 537-5
57 https://people.maths.bris.ac.uk/~majm/bib/steinhaus.pdf\n\n4. A. Haynes
and J. Marklof\, A five distance theorem for Kronecker sequences\, prepri
nt arXiv:2009.08444 https://people.maths.bris.ac.uk/~majm/bib/steinhaus2.p
df\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Stoll (University of Bayreuth)
DTSTART:20201126T160000Z
DTEND:20201126T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/58
DESCRIPTION:Title: An application of "Selmer group Chabauty" to arithmetic dynam
ics\nby Michael Stoll (University of Bayreuth) as part of Number Theor
y Web Seminar\n\n\nAbstract\nThe irreducibility or otherwise of iterates o
f polynomials is an\nimportant question in arithmetic dynamics. For exampl
e\, it is\nconjectured that whenever the second iterate of $x^2 + c$ (with
$c$ a\nrational number) is irreducible over $\\Q$\, then so are all itera
tes.\n\nA sufficient criterion for the iterates to be irreducible can be\n
expressed in terms of rational points on certain hyperelliptic curves.\nWe
will show how to use the "Selmer group Chabauty" method developed by\nthe
speaker to determine the set of rational points on a hyperelliptic\ncurve
of genus $7$. This leads to a proof that the seventh iterate of\n$x^2 + c
$ must be irreducible if the second iterate is. Assuming GRH\, we\ncan ext
end this to the tenth iterate.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adam Harper (University of Warwick)
DTSTART:20201215T100000Z
DTEND:20201215T110000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/59
DESCRIPTION:Title: Large fluctuations of random multiplicative functions\nby
Adam Harper (University of Warwick) as part of Number Theory Web Seminar\
n\n\nAbstract\nRandom multiplicative functions $f(n)$ are a well studied r
andom model for deterministic multiplicative functions like Dirichlet char
acters or the Mobius function. Arguably the first question ever studied ab
out them\, by Wintner in 1944\, was to obtain almost sure bounds for the l
argest fluctuations of their partial $\\sum_{n \\leq x} f(n)$\, seeking to
emulate the classical Law of the Iterated Logarithm for independent rando
m variables. It remains an open question to sharply determine the size of
these fluctuations\, and in this talk I will describe a new result in that
direction. I hope to get to some interesting details of the new proof in
the latter part of the talk\, but most of the discussion should be widely
accessible.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Lubotzky (Hebrew University of Jerusalem)
DTSTART:20201203T160000Z
DTEND:20201203T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/61
DESCRIPTION:Title: From Ramanujan graphs to Ramanujan complexes\nby Alexande
r Lubotzky (Hebrew University of Jerusalem) as part of Number Theory Web S
eminar\n\n\nAbstract\nRamanujan graphs are $k$-regular graphs with all no
n trivial eigenvalues bounded (in absolute value) by $2\\sqrt{k-1}$. They
are optimal expanders (from spectral point of view). Explicit constructio
ns of such graphs were given in the 80's as quotients of the Bruhat-Tits t
ree associated with $\\GL(2)$ over a local field $F$\, by the action of su
itable congruence subgroups of arithmetic groups. The spectral bound was p
roved using works of Hecke\, Deligne and Drinfeld on the "Ramanujan conjec
ture" in the theory of automorphic forms.\n\nThe work of Lafforgue\, exte
nding Drinfeld from $\\GL(2)$ to $\\GL(n)$\, opened the door for the cons
truction of Ramanujan complexes as quotients of the Bruhat-Tits buildings
associated with $\\GL(n)$ over $F$. This way one gets finite simplicial
complexes which on one hand are "random like" and at the same time have st
rong symmetries. These seemingly contradicting properties make them very u
seful for constructions of various external objects. \n\nRecently variou
s applications have been found in combinatorics\, coding theory and in rel
ation to Gromov's overlapping properties. We will survey some of these ap
plications.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jianya Liu (Shandong University)
DTSTART:20201222T100000Z
DTEND:20201222T110000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/62
DESCRIPTION:Title: Mobius disjointness for irregular flows\nby Jianya Liu (S
handong University) as part of Number Theory Web Seminar\n\n\nAbstract\nTh
e behavior of the Mobius function is central in the theory of prime number
s. A surprising connection with the theory of dynamical systems was discov
ered in 2010 by P. Sarnak\, who formulated the Mobius Disjointness Conject
ure (MDC)\, which asserts that the Mobius function is linearly disjoint fr
om any zero-entropy flows. This conjecture opened the way into a large bod
y of research on the interface of analytic number theory and ergodic theor
y. In this talk I will report how to establish MDC for a class of irregula
r flows\, which are in general mysterious and ill understood. This is base
d on joint works with P. Sarnak\, and with W. Huang and K. Wang.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jasmin Matz (University of Copenhagen)
DTSTART:20201124T100000Z
DTEND:20201124T110000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/63
DESCRIPTION:Title: Quantum ergodicity of compact quotients of $SL(n\,R)/SO(n)$ i
n the level aspect\nby Jasmin Matz (University of Copenhagen) as part
of Number Theory Web Seminar\n\n\nAbstract\nSuppose $M$ is a closed Rieman
nian manifold with an orthonormal basis $B$\nof $L^2(M)$ consisting of Lap
lace eigenfunctions. A classical result of\nShnirelman and others proves t
hat if the geodesic flow on the cotangent\nbundle of $M$ is ergodic\, then
$M$ is quantum ergodic\, in particular\, on\naverage\, the probability me
asures defined by the functions $f$ in $B$ on $M$\ntends on average toward
s the Riemannian measure on $M$ in the high\nenergy limit (i.e\, as the La
place eigenvalues of $f \\to \\infty$). \nWe now want to look at a level a
spect of this property\, namely\, instead\nof taking a fixed manifold and
high energy eigenfunctions\, we take a\nsequence of Benjamini-Schramm conv
ergent compact Riemannian manifolds\n$M_j$ together with Laplace eigenfunc
tions $f$ whose eigenvalue varies in\nshort intervals. This perspective ha
s been recently studied in the\ncontext of graphs by Anantharaman and Le M
asson\, and for hyperbolic\nsurfaces and manifolds by Abert\, Bergeron\, L
e Masson\, and Sahlsten. In\nmy talk I want to discuss joint work with F.
Brumley in which we study\nthis question in higher rank\, namely sequences
of compact quotients of\n$SL(n\,R)/SO(n)$.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gérald Tenenbaum (Université de Lorraine)
DTSTART:20201110T100000Z
DTEND:20201110T110000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/65
DESCRIPTION:Title: Recent progress on the Selberg-Delange method in analytic num
ber theory\nby Gérald Tenenbaum (Université de Lorraine) as part of
Number Theory Web Seminar\n\n\nAbstract\nLet $\\varrho$ be a complex numbe
r and let $f$ be a multiplicative arithmetic function whose Dirichlet seri
es takes the form $\\zeta(s)^\\varrho G(s)$\, where $\\zeta(s)$ is the Rie
mann zeta function and $G$ is associated to a multiplicative function $g$.
The classical Selberg-Delange method furnishes asymptotic estimates for t
he averages of $f$ under assumptions of either analytic continuation for $
G$\, or absolute convergence of a finite number of derivatives of $G(s)$ a
t $s=1$. We shall recall these statements and briefly describe the proofs.
The main part of of the lecture will be devoted to give an account on rec
ent works (in particular a joint paper with Régis de la Bretèche) consid
ering different set of hypotheses\, not directly comparable to the previou
s ones. We shall investigate what assumptions are sufficient to yield sha
rp asymptotic estimates for the averages of $f$.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jason Bell (University of Waterloo)
DTSTART:20201117T010000Z
DTEND:20201117T020000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/66
DESCRIPTION:Title: A transcendental dynamical degree\nby Jason Bell (Univers
ity of Waterloo) as part of Number Theory Web Seminar\n\n\nAbstract\nThe d
egree of a dominant rational map $f:\\mathbb{P}^n\\to \\mathbb{P}^n$ is th
e common degree of its homogeneous components. By considering iterates of
$f$\, one can form a sequence ${\\rm deg}(f^n)$\, which is submultiplicat
ive and hence has the property that there is some $\\lambda\\ge 1$ such th
at $({\\rm deg}(f^n))^{1/n}\\to \\lambda$. The quantity $\\lambda$ is cal
led the first dynamical degree of $f$. We’ll give an overview of the si
gnificance of the dynamical degree in complex dynamics and describe an exa
mple in which this dynamical degree is provably transcendental. This is j
oint work with Jeffrey Diller and Mattias Jonsson.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Imre Ruzsa (Alfréd Rényi Institute of Mathematics)
DTSTART:20210107T160000Z
DTEND:20210107T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/67
DESCRIPTION:Title: Additive decomposition of signed primes\nby Imre Ruzsa (A
lfréd Rényi Institute of Mathematics) as part of Number Theory Web Semin
ar\n\n\nAbstract\nAssuming the prime-tuple hypothesis\, the set of signed
primes is a sumset. More exactly\, there are infinite sets $A$\, $B$ of in
tegers such that $A+B$ consists exactly of the (positive and negative) pri
mes with $|p|>3$.\nI will also meditate on the possibility of a triple sum
and analogous problems for the set of squarefree numbers.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Sarnak (Institute for Advanced Study and Princeton Universit
y)
DTSTART:20210114T160000Z
DTEND:20210114T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/68
DESCRIPTION:Title: Summation formulae in spectral theory and number theory (A ta
lk in honor of Zeev Rudnick's 60th Birthday)\nby Peter Sarnak (Institu
te for Advanced Study and Princeton University) as part of Number Theory W
eb Seminar\n\n\nAbstract\nThe Poisson Summation formula\, Riemann-Guinand-
Weil explicit formula\, Selberg Trace Formula and Lefschetz Trace formula
in the function field\, are starting points for a number of Zeev Rudnick's
works. We will review some of these before describing some recent applica
tions (joint with P. Kurasov) of Lang's $\\mathbb{G}_m$ conjectures to the
additive structure of the spectra of metric graphs and crystalline measur
es.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lior Bary-Soroker (Tel Aviv University)
DTSTART:20210121T160000Z
DTEND:20210121T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/69
DESCRIPTION:Title: Random Polynomials\, Probabilistic Galois Theory\, and Finit
e Field Arithmetic\nby Lior Bary-Soroker (Tel Aviv University) as par
t of Number Theory Web Seminar\n\n\nAbstract\nAbstract: In the talk we wil
l discuss recent advances on the following two questions: \n\nLet $A(X) =
\\sum \\pm X^i$ be a random polynomial of degree $n$ with coefficients tak
ing the values $-1\,1$ independently each with probability $1/2$.\n\nQ1:
What is the probability that $A$ is irreducible as the degree goes to infi
nity?\n\nQ2: What is the typical Galois group of $A$?\n\nOne believes that
the answers are YES and THE FULL SYMMETRIC GROUP\, respectively. These qu
estions were studied extensively in recent years\, and we will survey the
tools developed to attack these problems and partial results.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:William Banks (University of Missouri)
DTSTART:20210128T160000Z
DTEND:20210128T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/70
DESCRIPTION:Title: On the distribution of reduced fractions with squarefree deno
minators\nby William Banks (University of Missouri) as part of Number
Theory Web Seminar\n\n\nAbstract\nAbstract: In this talk we discuss how th
e nonvanishing of the Riemann zeta function in a half-plane $\\{\\sigma>\\
sigma_0\\}$\, with some real $\\sigma_0<1$\, is equivalent to a strong st
atement about the distribution in the unit interval of reduced fractions w
ith squarefree denominators.\n\nThe approach utilizes an unconditional gen
eralization of a theorem of Blomer concerning the distribution "on averag
e" of squarefree integers in arithmetic progressions to large moduli.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleksiy Klurman (University of Bristol)
DTSTART:20210204T160000Z
DTEND:20210204T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/71
DESCRIPTION:Title: On the zeros of Fekete polynomials\nby Oleksiy Klurman (U
niversity of Bristol) as part of Number Theory Web Seminar\n\n\nAbstract\n
Since their discovery by Dirichlet in the nineteenth century\, Fekete poly
nomials (with coefficients being Legendre symbols) and their zeros attract
ed considerable attention\, in particular\, due to their intimate connecti
on with putative Siegel zero and small class number problem. The goal of t
his talk is to discuss what we knew\, know and would like to know about ze
ros of such (and related) polynomials. Joint work with Y. Lamzouri and M.
Munsch.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Don Zagier (Max Planck Institute for Mathematics)
DTSTART:20210211T160000Z
DTEND:20210211T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/72
DESCRIPTION:Title: Analytic functions related to zeta-values\, cotangent product
s\, and the cohomology of $SL_2(\\Z)$\nby Don Zagier (Max Planck Insti
tute for Mathematics) as part of Number Theory Web Seminar\n\n\nAbstract\n
I will report on the properties of various functions\, going back essentia
lly to Herglotz\, that relate to a number of different topics in number th
eory\, including those in the title but also others like Hecke operators o
r Stark's conjectures. This is joint work with Danylo Radchenko.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gabriel Dill (University of Oxford)
DTSTART:20210218T160000Z
DTEND:20210218T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/73
DESCRIPTION:Title: Unlikely Intersections and Distinguished Categories\nby G
abriel Dill (University of Oxford) as part of Number Theory Web Seminar\n\
n\nAbstract\nAfter a general introduction to the field of unlikely interse
ctions\, I present current work in progress with Fabrizio Barroero\, in wh
ich we propose an axiomatic approach towards studying unlikely intersectio
ns by introducing the framework of distinguished categories. This includes
commutative algebraic groups and mixed Shimura varieties. It allows to us
to define all basic concepts of the field and prove some fundamental fact
s about them\, e.g. the defect condition. In some categories that we call
very distinguished\, we are able to show some implications between Zilber-
Pink statements with respect to base change. This also yields new uncondit
ional results on the Zilber-Pink conjecture for curves in various contexts
.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sanju Velani (University of York)
DTSTART:20210304T200000Z
DTEND:20210304T210000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/74
DESCRIPTION:Title: The Shrinking Target Problem for Matrix Transformations of To
ri\nby Sanju Velani (University of York) as part of Number Theory Web
Seminar\n\n\nAbstract\nLet $T$ be a $d\\times d$ matrix with integral coef
ficients.\nThen $T$ determines a self-map of the $d$-dimensional torus $\\
mathbb{T}^d=\\R^d/\\Z^d$.\nChoose for each natural number $n$ a ball $B(n)
$ in $X$\n and suppose that $B(n+1)$ has smaller radius than $B(n)$ for al
l $n$.\nThus the ball shrinks as $n$ increases. \nNow let $W$ be the set o
f points $x\\in \\mathbb{T}^d$ such that\n $T^n(x)\\in B(n)$ for infinitel
y many $n\\in\n$. The size of $W$ measured in terms of $d$-dimensional Leb
esgue measure (restricted to $\\mathbb{T}^d$) and Haudsorff dimension are
pretty much well understood. \n In this talk I explore the situation in w
hich the points $ x \\in \\mathbb{T}^d$ are restricted to a nice subset
${\\mathcal M}$ (such as an analytic sub-manifold) of $\\mathbb{T}^d$\; th
at is\, the points of interest are functionally dependent. I will essenti
ally concentrate on the situation when $d=2$\, $T$ has first row $(2\,0)
$ and second row $(0\,3)$\n and ${\\mathcal M}$ is the diagonal. In this
special case\, given a decreasing function $\\psi$\, understanding the
shrinking target set $W \\cap {\\mathcal M}$ is equivalent to understandi
ng the set of $x\\in [0\,1]$ such that $ \\max\\{\\|2^nx\\|\, \\|3^nx\\|\\
}<\\psi(n) $ for infinitely many $n\\in\n$. \n \n\n \n This is joint work
with Bing Li (South China University of Technology)\, Lingmin Liao (UPEC)
and Evgeniy Zorin (York).\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chantal David (Concordia University)
DTSTART:20210311T200000Z
DTEND:20210311T210000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/76
DESCRIPTION:Title: Moments and non-vanishing of cubic Dirichlet $L$-functions at
$s=\\frac{1}{2}$\nby Chantal David (Concordia University) as part of
Number Theory Web Seminar\n\n\nAbstract\nA famous conjecture of Chowla pre
dicts that $L(\\frac{1}{2}\,\\chi)\\ne 0$ for all Dirichlet $L$-functions\
nattached to primitive characters $\\chi$. It was conjectured first in the
case where $\\chi$ is a quadratic\ncharacter\, which is the most studied
case. For quadratic Dirichlet $L$-functions\, Soundararajan\nproved that a
t least 87.5% of the quadratic Dirichlet $L$-functions do not vanish at $s
=\\frac{1}{2}$.\nUnder GRH\, there are slightly stronger results by Ozlek
and Snyder.\n\nWe present in this talk the first result showing a positive
proportion of cubic Dirichlet\n$L$-functions non-vanishing at $s=\\frac{1
}{2}$ for the non-Kummer case over function fields. This can\nbe achieved
by using the recent breakthrough work on sharp upper bounds for moments of
\nSoundararajan\, Harper and Lester-Radziwill. Our results would transfer
over number fields\,\nbut we would need to assume GRH in this case.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shabnam Akhtari (University of Oregon)
DTSTART:20210318T200000Z
DTEND:20210318T210000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/77
DESCRIPTION:Title: Orders in Quartic Number Fields and Classical Diophantine Equ
ations\nby Shabnam Akhtari (University of Oregon) as part of Number Th
eory Web Seminar\n\n\nAbstract\nAn order $\\mathcal{O}$ in an algebraic nu
mber field is called monogenic if over $\\mathbb{Z}$ it can be generated b
y one element. Gy\\H{o}ry has shown that there are finitely equivalence cl
asses \n$\\alpha \\in \\mathcal{O}$ such that $\\mathcal{O} = \\mathbb{Z}[
\\alpha]$\, where two algebraic integers $\\alpha$ and $\\alpha'$ are call
ed equivalent if $\\alpha + \\alpha'$ or $\\alpha - \\alpha'$ is a ration
al integer. An interesting problem is to count the number of monogenizati
ons of a given monogenic order. First we will note\, for a given order $\\
mathcal{O}$\, that \n$$\n\\mathcal{O} = \\mathbb{Z}[\\alpha] \\\, \\quad \
\textrm{in} \\\, \\\, \\alpha\,\n$$\nis indeed a Diophantine equation. The
n we will modify some old algorithmic results to obtain new and improved u
pper bounds for the number of monogenizations of a quartic order.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vitaly Bergelson (Ohio State University)
DTSTART:20210325T203000Z
DTEND:20210325T213000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/78
DESCRIPTION:Title: A "soft" dynamical approach to the Prime Number Theorem and d
isjointness of additive and multiplicative semigroup actions\nby Vital
y Bergelson (Ohio State University) as part of Number Theory Web Seminar\n
\n\nAbstract\nWe will discuss a new type of ergodic theorem which has amon
g its corollaries numerous classical results from multiplicative number th
eory\, including the Prime Number Theorem\, a theorem of Pillai-Selberg an
d a theorem of Erdős-Delange. This ergodic approach leads to a new dynami
cal framework for a general form of Sarnak’s Möbius disjointness conjec
ture which focuses on the "joint independence" of actions of $(\n\,+)$ and
$(\n\,×)$. The talk is based on recent joint work with Florian Richter.\
n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Adamczewski (Université Claude Bernard Lyon 1)
DTSTART:20210401T150000Z
DTEND:20210401T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/79
DESCRIPTION:Title: Furstenberg's conjecture\, Mahler's method\, and finite autom
ata\nby Boris Adamczewski (Université Claude Bernard Lyon 1) as part
of Number Theory Web Seminar\n\n\nAbstract\nIt is commonly expected that e
xpansions of numbers in multiplicatively independent bases\, such as 2 and
10\, should have no common structure. However\, it seems extraordinarily
difficult to confirm this naive heuristic principle in some way or another
. In the late 1960s\, Furstenberg suggested a series of conjectures\, whic
h became famous and aim to capture this heuristic. The work I will discuss
in this talk is motivated by one of these conjectures. Despite recent rem
arkable progress by Shmerkin and Wu\, it remains totally out of reach of t
he current methods. While Furstenberg’s conjectures take place in a dyna
mical setting\, I will use instead the language of automata theory to form
ulate some related problems that formalize and express in a different way
the same general heuristic. I will explain how the latter can be solved th
anks to some recent advances in Mahler’s method\; a method in transcende
ntal number theory initiated by Mahler at the end of the 1920s. This a joi
nt work with Colin Faverjon.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:János Pintz (Alfréd Rényi Institute of Mathematics)
DTSTART:20210408T150000Z
DTEND:20210408T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/80
DESCRIPTION:Title: On the mean value of the remainder term of the prime number f
ormula\nby János Pintz (Alfréd Rényi Institute of Mathematics) as p
art of Number Theory Web Seminar\n\n\nAbstract\nThere are several methods
to obtain a lower bound for the mean value of the absolute value of the re
mainder term of the prime number formula as function of a hypothetical zer
o of the Riemann Zeta function off the critical line. (The case when the R
iemann Hypothesis is true can be treated easier.) The most efficient ones
include results of Knapowski-Turán\, Sz. Gy. Révész \, and the author\,
proved by several different methods\n\nThe result to be proved in the lec
ture provides (again with an other method) a quite good lower bound and it
has the good feature (which is useful in further applications too) that i
nstead of the whole interval $[0\,X]$ it gives a good lower bound for the
average on $[F(X)\, X]$ with $\\log F(X)$ close to $\\log X$ (that is on "
short" intervals measured with the logarithmic scale).\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Keating (University of Oxford)
DTSTART:20210415T150000Z
DTEND:20210415T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/81
DESCRIPTION:Title: Joint Moments\nby Jonathan Keating (University of Oxford)
as part of Number Theory Web Seminar\n\n\nAbstract\nI will discuss the jo
int moments of the Riemann zeta-function and its derivative\, and the corr
esponding joint moments of the characteristic polynomials of random unitar
y matrices and their derivatives.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Akshay Venkatesh (Institute for Advanced Study)
DTSTART:20210506T150000Z
DTEND:20210506T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/82
DESCRIPTION:Title: A brief history of Hecke operators\nby Akshay Venkatesh (
Institute for Advanced Study) as part of Number Theory Web Seminar\n\n\nAb
stract\nThis is an expository lecture about Hecke operators\, in the conte
xt of number theory. We will trace some of the history of the ideas\, sta
rting before Hecke's birth and proceeding through the subsequent century.
In particular we will discuss some of the original motivations and then th
e impact of ideas from representation theory and algebraic geometry. This
lecture is aimed at non-experts.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Kontorovich (Rutgers University)
DTSTART:20210513T150000Z
DTEND:20210513T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/83
DESCRIPTION:Title: Arithmetic Groups and Sphere Packings\nby Alex Kontorovic
h (Rutgers University) as part of Number Theory Web Seminar\n\n\nAbstract\
nWe discuss recent progress on understanding connections between the objec
ts in the title.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pietro Corvaja (University of Udine)
DTSTART:20210429T150000Z
DTEND:20210429T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/84
DESCRIPTION:Title: On the local-to-global principle for value sets\nby Pietr
o Corvaja (University of Udine) as part of Number Theory Web Seminar\n\n\n
Abstract\nGiven a finite morphism $f: X \\to Y$ between algebraic curves o
ver number fields\, we study the set of rational (or integral) points in $
Y$ having a pre-image in every $p$-adic completion of the number field\, b
ut no rational pre-images. In particular\, we investigate whether this set
can be infinite.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Tichy (TU Graz)
DTSTART:20210527T150000Z
DTEND:20210527T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/85
DESCRIPTION:Title: Equidistribution\, exponential sums and van der Corput sets\nby Robert Tichy (TU Graz) as part of Number Theory Web Seminar\n\n\nAb
stract\nThe talk starts with a survey on Sarkoezy`s results on difference
sets and with Furstenberg`s dynamic approach to additive problems. We pres
ent some results of a joint work with Bergelson\, Kolesnik\, Son and Madri
tsch concerning multidimensional van der Corput sets based on new bounds f
or exponential sums. In a second part we give a brief introduction on equi
distribution theory focusing on the interplay of exponential sums with dif
ference theorems. In a third part Hardy fields are discussed in some detai
l. This concept was introduced to equidistribution theory by Boshernitzan
and it tuned out to be very fruitful. We will report on recent results of
Bergelson et al. and at the very end on applications to diophantine appro
ximation. This includes results concerning the approximation of polynomial
-like functions along primes which were established in a joint work with M
adritsch and sharpened very recently by my PhD student Minelli.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Renate Scheidler (University of Calgary)
DTSTART:20210422T150000Z
DTEND:20210422T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/86
DESCRIPTION:Title: Computing modular polynomials and isogeny graphs of rank $2$
Drinfeld modules\nby Renate Scheidler (University of Calgary) as part
of Number Theory Web Seminar\n\n\nAbstract\nDrinfeld modules represent the
function field analogue of the theory of complex multiplication. They wer
e introduced as "elliptic modules" by Vladimir Drinfeld in the 1970s in th
e course of proving the Langlands conjectures for $\\GL(2)$ over global fu
nction fields. Drinfeld modules of rank $2$ exhibit very similar behaviour
to elliptic curves: they are classified as ordinary or supersingular\, su
pport isogenies and their duals\, and their endomorphism rings have an ana
logous structure. Their isomorphism classes are parameterized by $j$-invar
iants\, and Drinfeld modular polynomials can be used to compute their isog
eny graphs whose ordinary connected components take the shape of volcanos.
While the rich analytic and algebraic theory of Drinfeld modules has unde
rgone extensive investigation\, very little has been explored from a compu
tational perspective. This research represents the first foray in this dir
ection\, introducing an algorithm for computing Drinfeld modular polynomia
ls and isogeny graphs. \n\nThis is joint work with Perlas Caranay and Matt
Greenberg\, as well as ongoing research with Edgar Pacheco Castan. Some f
amiliarity with elliptic curves is expected for this talk\, but no prior k
nowledge of Drinfeld modules is assumed.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shou-Wu Zhang (Princeton University)
DTSTART:20210617T150000Z
DTEND:20210617T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/87
DESCRIPTION:Title: Adelic line bundles over quasi-projective varieties\nby S
hou-Wu Zhang (Princeton University) as part of Number Theory Web Seminar\n
\n\nAbstract\nFor quasi-projective varieties over finitely generated field
s\, we develop a theory of adelic line bundles including an equidistributi
on theorem for Galois orbits of small points. In this lecture\, we will ex
plain this theory and its application to arithmetic of abelian varieties\,
dynamical systems\, and their moduli. This is a joint work with Xinyi Yua
n.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Antoine Chambert-Loir (Université Paris-Diderot)
DTSTART:20210603T180000Z
DTEND:20210603T190000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/88
DESCRIPTION:Title: From complex function theory to non-archimedean spaces - a nu
mber theoretical thread\nby Antoine Chambert-Loir (Université Paris-D
iderot) as part of Number Theory Web Seminar\n\n\nAbstract\nDiophantine ge
ometry and complex function theory have a long and well known history of m
utual friendship\, attested\, for example\, by the fruitful interactions b
etween height functions and potential theory. In the last 50 years\, inter
actions even deepened with the invention of Arakelov geometry (Arakelov\,
Gillet/Soulé\, Faltings) and its application by Szpiro/Ullmo/Zhang to equ
idistribution theorems and the Bogomolov conjecture. Roughly at the same t
ime\, Berkovich invented a new kind of non-archimedean analytic spaces whi
ch possess a rich\nand well behaved geometric structure. This opened the w
ay to non-archimedean potential theory (Baker/Rumely\, Favre/Rivera-Leteli
er)\, or to arithmetic/geometric equidistribution theorems in this case. M
ore recently\, Ducros and myself introduced basic ideas from tropical geom
etry and a construction of Lagerberg to construct a calculus of $(p\,q)$-f
orms on Berkovich spaces\, which is an analogue of the corresponding calcu
lus on complex manifolds\, and seems to be an attractive candidate for bei
ng the $p$-adic side of height function theory.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alice Silverberg (University of California\, Irvine)
DTSTART:20210520T150000Z
DTEND:20210520T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/89
DESCRIPTION:Title: Cryptographic Multilinear Maps and Miscellaneous Musings\
nby Alice Silverberg (University of California\, Irvine) as part of Number
Theory Web Seminar\n\n\nAbstract\nRecognizing that many of us have Zoom f
atigue\, I will keep this talk light\, without too many technical details.
In addition to discussing an open problem on multilinear maps that has ap
plications to cryptography\, I'll give miscellaneous musings about things
I've learned over the years that I wish I'd learned sooner.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Annette Huber-Klawitter (University of Freiburg)
DTSTART:20210624T150000Z
DTEND:20210624T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/90
DESCRIPTION:Title: Periods and O-minimality\nby Annette Huber-Klawitter (Uni
versity of Freiburg) as part of Number Theory Web Seminar\n\n\nAbstract\nR
oughly\, periods are numbers obtained by integrating algebraic\ndifferenti
al forms over domains of integration also of arithmetic\nnature. I am goi
ng to give a survey on the state of the period\nconjecture and different p
oints of view. I also want to present a\nrelation to o-minimal geometry.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/90/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthew Young (Texas A&M University)
DTSTART:20210610T150000Z
DTEND:20210610T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/91
DESCRIPTION:Title: The Weyl bound for Dirichlet L-functions\nby Matthew Youn
g (Texas A&M University) as part of Number Theory Web Seminar\n\n\nAbstrac
t\nThere is an analogy between the behavior of the Riemann zeta function h
igh in the critical strip\, and the behavior of Dirichlet $L$-functions of
large conductors. In many important problems\, our understanding of Diri
chlet $L$-functions is weaker than for zeta\; for example\, the zero-free
regions are not of the same quality due to the possible Landau-Siegel zero
. This talk will discuss recent progress (joint with Ian Petrow) on subco
nvexity bounds for Dirichlet $L$-functions. These new bounds now match the
original subconvexity bound for the zeta function derived by Hardy and Li
ttlewood using Weyl's differencing method.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ben Green
DTSTART:20210225T160000Z
DTEND:20210225T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/92
DESCRIPTION:Title: New lower bounds for van der Waerden numbers\nby Ben Gree
n as part of Number Theory Web Seminar\n\n\nAbstract\nColour $\\{1\,..\,N\
\}$ red and blue\, in such a manner that no $3$ of the blue elements are i
n arithmetic progression. How long an arithmetic progression of red elemen
ts must there be? It had been speculated based on numerical evidence that
there must always be a red progression of length about $\\sqrt{N}$. I will
describe a construction which shows that this is not the case - in fact\,
there is a colouring with no red progression of length more than about $\
\exp ((\\log N)^{3/4})$\, and in particular less than any fixed power of $
N$.\n\nI will give a general overview of this kind of problem (which can b
e formulated in terms of finding lower bounds for so-called van der Waerde
n numbers)\, and an overview of the construction and some of the ingredien
ts which enter into the proof. The collection of techniques brought to bea
r on the problem is quite extensive and includes tools from diophantine ap
proximation\, additive number theory and\, at one point\, random matrix th
eory.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manjul Bhargava (Princeton University)
DTSTART:20210701T150000Z
DTEND:20210701T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/93
DESCRIPTION:Title: Galois groups of random integer polynomials (A talk in honor
of Don Zagier's 70th birthday)\nby Manjul Bhargava (Princeton Universi
ty) as part of Number Theory Web Seminar\n\n\nAbstract\nOf the $(2H+1)^n$
monic integer polynomials $f(x)=x^n+a_1 x^{n-1}+\\cdots+a_n$ with $\\max\\
{|a_1|\,\\ldots\,|a_n|\\}\\leq H$\, how many have associated Galois group
that is not the full symmetric group $S_n$? There are clearly $\\gg H^{n-1
}$ such polynomials\, as can be seen by setting $a_n=0$. In 1936\, van der
Waerden conjectured that $O(H^{n-1})$ should in fact also be the correct
upper bound for the count of such polynomials. The conjecture has been kno
wn for $n\\leq 4$ due to work of van der Waerden and Chow and Dietmann. I
n this talk\, we prove the "Weak van der Waerden Conjecture"\, which state
s that the number of such polynomials is $O_\\epsilon(H^{n-1+\\epsilon})$\
, for all degrees $n$.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brian Conrey (American Institute of Mathematics)
DTSTART:20210708T150000Z
DTEND:20210708T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/94
DESCRIPTION:Title: Moments\, ratios\, arithmetic functions in short intervals an
d random matrix averages\nby Brian Conrey (American Institute of Mathe
matics) as part of Number Theory Web Seminar\n\n\nAbstract\nWe discuss how
the conjectures for moments of $L$-functions\nimply short interval averag
es of the $L$-coefficient convolutions. Similarly\nthe ratios conjectures
lead to short interval averages of the convolutions\nof coefficients at al
most primes. These in turn are related to random matrix averages considere
d by Diaconis - Gamburd and by Diaconis - Shahshahani.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ricardo Menares (Pontificia Universidad Católica de Chile)
DTSTART:20210715T150000Z
DTEND:20210715T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/95
DESCRIPTION:Title: $p$-adic distribution of CM points\nby Ricardo Menares (P
ontificia Universidad Católica de Chile) as part of Number Theory Web Sem
inar\n\n\nAbstract\nCM points are the isomorphism classes of CM elliptic c
urves. When ordered by the absolute value of the discriminant of the endom
orphism ring\, CM points are distributed along the complex (level one) mod
ular curve according to the hyperbolic measure. This statement was proved
by Duke for fundamental discriminants and later\, building on this work\,
Clozel and Ullmo proved it in full generality.\n\nIn this talk\, we establ
ish the $p$-adic analogue of this result. Namely\, for a fixed prime $p$ w
e regard the CM points as a subset of the $p$-adic space attached to the m
odular curve and we classify the possible accumulation measures of CM poin
ts as the discriminant varies. In particular\, we find that there are infi
nitely many such measures. This is in stark contrast to the complex case\,
where the hyperbolic measure is the unique accumulation measure. \n\nAs a
n application\, we show that for any finite set $S$ of prime numbers\, the
set of singular moduli which are $S$-units is finite.\n\nThis is joint wo
rk with Sebastián Herrero (PUC Valparaíso) and Juan Rivera-Letelier (Roc
hester).\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/95/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lars Kühne (University of Copenhagen)
DTSTART:20210902T150000Z
DTEND:20210902T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/96
DESCRIPTION:Title: The uniform Bogomolov conjecture for algebraic curves\nby
Lars Kühne (University of Copenhagen) as part of Number Theory Web Semin
ar\n\n\nAbstract\nI will present an equidistribution result for families o
f (non-degenerate) subvarieties in a (general) family of abelian varieties
. This extends a result of DeMarco and Mavraki for curves in fibered produ
cts of elliptic surfaces\, but it also follows from independent work by Yu
an and Zhang\, which has been recently reported in this seminar. I will th
erefore focus on the application that motivated my work\, namely a uniform
version of the classical Bogomolov conjecture for curves embedded in thei
r Jacobians. This has been previously only known in a few select cases by
work of David–Philippon and DeMarco–Krieger–Ye. Furthermore\, one ca
n deduce a rather uniform version of the Mordell-Lang conjecture by comple
menting a result of Dimitrov–Gao–Habegger: The number of rational poin
ts on a smooth algebraic curve defined over a number field can be bounded
solely in terms of its genus and the Mordell-Weil rank of its Jacobian. Ag
ain\, this was previously known only under additional assumptions (Stoll\,
Katz–Rabinoff–Zureick-Brown). All these results have been recently ge
neralized beyond curves in joint work with Ziyang Gao and Tangli Ge\, but
I will restrict to the case of curves for simplicity.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/96/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean-Marc Deshouillers (Institut de Mathématiques de Bordeaux)
DTSTART:20211014T150000Z
DTEND:20211014T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/97
DESCRIPTION:Title: Are factorials sums of three cubes?\nby Jean-Marc Deshoui
llers (Institut de Mathématiques de Bordeaux) as part of Number Theory We
b Seminar\n\n\nAbstract\nLet $\\mathcal{C}_3$ be the set of non-negative
integers which are sums of the cubes of three non-negative integers and le
t $C_3$ be their counting function\, id est\n$$\nC_3(x)= \\operatorname{C
ard}\\{n \\le x \\colon n \\in \\mathcal{C}_3\\}.\n$$\nOur knowledge of su
ms of three cubes is somewhat limited\, for example\, we do not know wheth
er there exists a positive real $c$ such that for any sufficiently large $
x$ one has\n$$\nC_3(x) \\ge cx.\n$$\nNumerical and probabilistic results a
re in favour of \n$$\nC_3(x) \\sim cx\, \\text{ where } c=0.0999425... \\t
ext{ as $x$ tends to infinity}.\n$$\n\nNumerical results presented in the
chapter A267414 of the OEIS project suggest that factorials are very ofte
n sums of three cubes and even that as soon as $n$ is large enough\, $n!$
is a sum of three cubes. The aim of the talk is to present a probability
model\, consistent with the actual distribution of cubes\, in which\, almo
st surely\, as soon as $n$ is large enough\, $n!$ is a sum of three pseudo
-cubes.\n\nWe shall also give two applications of our result to classical
problems on sums of cubes. \nThe result presented in the talk have been j
ointly obtained with Altug Alkan (Istanbul)\, François Hennecart (Saint
Étienne) et Bernard Landreau (Angers).\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/97/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martin Sombra (ICREA and University of Barcelona)
DTSTART:20210916T150000Z
DTEND:20210916T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/98
DESCRIPTION:Title: The mean height of the solution set of a system of polynomial
equations\nby Martin Sombra (ICREA and University of Barcelona) as pa
rt of Number Theory Web Seminar\n\n\nAbstract\nBernstein’s theorem allow
s to predict the number of solutions of a system of Laurent\npolynomial eq
uations in terms of combinatorial invariants. When the coefficients of the
system\nare algebraic numbers\, we can ask about the height of these solu
tions. Based on an on-going project with Roberto Gualdi (Regensburg)\, I w
ill explain how one can approach this question using tools from the Arakel
ov geometry of toric varieties.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/98/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dimitris Koukoulopoulos (University of Montreal)
DTSTART:20211028T150000Z
DTEND:20211028T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/99
DESCRIPTION:Title: Towards a high-dimensional theory of divisors of integers
\nby Dimitris Koukoulopoulos (University of Montreal) as part of Number Th
eory Web Seminar\n\n\nAbstract\nIn this talk\, I will survey some results
about high-dimensional phenomena in the theory of divisors of integers. \n
\nFix an integer $k\\ge2$ and pick an integer $n\\le x$ uniformly at rando
m. We then consider the following two basic problems:\nWhat are the chance
s that $n$ can be factored as $n=d_1\\cdots d_k$ with each factor $d_i$ ly
ing in some prescribed dyadic interval $[y_i\,2y_i]$?\nWhat are the chance
s that we can find $k$ divisors of $n$\, say $d_1\,\\dots\,d_k$\, such tha
t $|\\log(d_j/d_i)|<1$ for all $i\,j$\, and which are all composed from a
prescribed set of prime factors of $n$?\nThe first problem is a high-dimen
sional generalization of the Erdős multiplication table problem\; it is w
ell-understood when $k\\le 6$\, but less so when $k\\ge7$. The second prob
lem is related to Hooley’s function $\\Delta(n):=\\max_u \\#\\{d|n:u<\\l
og d\\le u+1\\}$ that measures the concentration of the sequence of diviso
rs of $n$\, and that has surprising applications to Diophantine number the
ory.\n\nIn recent work with Kevin Ford and Ben Green\, we built on the ear
lier work on Problem 1 to develop a new approach to Problem 2. This led to
an improved lower bound on the almost-sure behaviour of Hooley’s $\\Del
ta$-function\, that we conjecture to be optimal. The new ideas might in tu
rn shed light to Problem 1 and other high-dimensional phenomena about divi
sors of integers.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/99/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arno Fehm (Technische Universität Dresden)
DTSTART:20210729T150000Z
DTEND:20210729T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/100
DESCRIPTION:Title: Is $\\Z$ diophantine in $\\Q$?\nby Arno Fehm (Technische
Universität Dresden) as part of Number Theory Web Seminar\n\n\nAbstract\
nAre the integers the projection of the rational zeros of a polynomial in
several variables onto the first coordinate? The aim of this talk is to mo
tivate and discuss this longstanding question. I will survey some results
regarding diophantine sets and Hilbert's tenth problem (the existence of a
n algorithm that decides whether a polynomial has a zero) in fields and wi
ll discuss a few conjectures\, some classical and some more recent\, that
suggest that the answer to the question should be negative.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/100/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandru Zaharescu (University of Illinois at Urbana-Champaign)
DTSTART:20210826T210000Z
DTEND:20210826T220000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/101
DESCRIPTION:Title: Some remarks on Landau - Siegel zeros\nby Alexandru Zaha
rescu (University of Illinois at Urbana-Champaign) as part of Number Theor
y Web Seminar\n\n\nAbstract\nIn the first part of the talk I will survey s
ome known results related to the hypothetical existence of Landau - Siegel
zeros. In the second part of the talk I will discuss some recent joint wo
rk with Hung Bui and Kyle Pratt in which we show that the existence of Lan
dau - Siegel zeros has implications for the behavior of $L$ - functions at
the central point.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/101/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frank Calegari (University of Chicago)
DTSTART:20210805T150000Z
DTEND:20210805T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/102
DESCRIPTION:Title: Digits\nby Frank Calegari (University of Chicago) as par
t of Number Theory Web Seminar\n\n\nAbstract\nWe discuss some results conc
erning the decimal expansion of $1/p$ for primes $p$\, some due to Gauss\,
and some from the present day. This is work in progress with Soundararaja
n which we may well write up one day.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/102/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kumar Murty (University of Toronto)
DTSTART:20210722T150000Z
DTEND:20210722T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/103
DESCRIPTION:Title: Periods and Mixed Motives\nby Kumar Murty (University of
Toronto) as part of Number Theory Web Seminar\n\n\nAbstract\nWe discuss s
ome consequences of Grothendieck's Period Conjecture in the context of mix
ed motives. In particular\, this conjecture implies that $\\zeta(3)$\, $\\
log 2$ and $\\pi$ are algebraically independent (contrary to an expectatio
n of Euler). After some 'motivation' and introductory remarks on periods\,
we derive our consequences as a result of studying mixed motives whose Ga
lois group has a large unipotent radical. This is joint work with Payman E
skandari.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/103/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesco Amoroso (University of Caen)
DTSTART:20210812T150000Z
DTEND:20210812T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/104
DESCRIPTION:Title: Bounded Height in Pencils of Subgroups of finite rank\nb
y Francesco Amoroso (University of Caen) as part of Number Theory Web Semi
nar\n\n\nAbstract\n[Joint work with D. Masser and U. Zannier] \n\nLet $n>1
$ be a varying natural number. By a result of Beukers\, the solutions of $
t^n+(1-t)^n=1$ have uniformly bounded height. What happens if we allow rat
ional exponents? \n\nWe consider the analogous question replacing the affi
ne curve $x+y=1$ with an arbitrary irreducible curve and $\\{t^n | n \\tex
trm{ rational}\\}$ with the division group of a finitely generated subgrou
p.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/104/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emmanuel Kowalski (ETH Zürich)
DTSTART:20210909T150000Z
DTEND:20210909T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/105
DESCRIPTION:Title: Harmonic analysis over finite fields and equidistribution\nby Emmanuel Kowalski (ETH Zürich) as part of Number Theory Web Seminar
\n\n\nAbstract\nIn 1976\, Deligne defined a geometric version of the Fouri
er transform over finite fields\, leading to significant applications in n
umber theory.\n\nFor a number of applications\, including equidistribution
of exponential sums parameterized by multiplicative characters\, it would
be very helpful to have a similar geometric harmonic analysis for other g
roups. I will discuss ongoing joint work with A. Forey and J. Fresán in w
hich we establish some results in this direction by generalizing ideas of
Katz. I will present the general equidistribution theorem for exponential
sums parameterized by characters that we obtain\, and discuss applications
\, as well as challenges\, open questions and mysteries.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/105/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anish Ghosh (Tata Institute of Fundamental Research)
DTSTART:20210930T150000Z
DTEND:20210930T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/106
DESCRIPTION:Title: Values of quadratic forms at integer points\nby Anish Gh
osh (Tata Institute of Fundamental Research) as part of Number Theory Web
Seminar\n\n\nAbstract\nA famous theorem of Margulis\, resolving a conjectu
re of Oppenheim\, states that an indefinite\, irrational quadratic form in
at least three variables takes a dense set of values at integer points. R
ecently there has been a push towards establishing effective versions of M
argulis's theorem. I will explain Margulis's approach to this problem whic
h involves the ergodic theory of group actions on homogeneous spaces. I wi
ll then discuss some new effective results in this direction. These result
s use a variety of techniques including tools from ergodic theory\, analyt
ic number theory as well as the geometry of numbers.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/106/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alina Carmen Cojocaru (University of Illinois at Chicago and Insti
tute of Mathematics of the Romanian Academy)
DTSTART:20210923T150000Z
DTEND:20210923T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/107
DESCRIPTION:Title: Bounds for the distribution of the Frobenius traces associat
ed to abelian varieties\nby Alina Carmen Cojocaru (University of Illin
ois at Chicago and Institute of Mathematics of the Romanian Academy) as pa
rt of Number Theory Web Seminar\n\n\nAbstract\nIn 1976\, Serge Lang and Ha
le Trotter conjectured the asymptotic growth of the number $\\pi_A(x\, t)$
of primes $p < x$ for which the Frobenius trace $a_p$ of a non-CM ellipti
c curve $A/\\mathbb{Q}$ equals an integer $t$. Even though their conjectur
e remains open\, over the past decades the study of the counting function
$\\pi_A(x\, t)$ has witnessed remarkable advances. We will discuss general
izations of such studies in the setting of an abelian variety $A/\\mathbb{
Q}$ of arbitrary dimension and we will present non-trivial upper bounds fo
r the corresponding counting function $\\pi_A(x\, t)$. This is joint work
with Tian Wang (University of Illinois at Chicago).\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/107/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Henryk Iwaniec (Rutgers University)
DTSTART:20211007T150000Z
DTEND:20211007T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/108
DESCRIPTION:Title: Remarks on the large sieve (A talk in honor of John Friedlan
der's 80th birthday)\nby Henryk Iwaniec (Rutgers University) as part o
f Number Theory Web Seminar\n\n\nAbstract\nThe concept of the large sieve
will be discussed in various contexts. The power and limitation of basic e
stimates will be illustrated with some examples. Recent work on the large
sieve for characters to prime moduli will be explained.\n\nSpecial Chairs:
Leo Goldmakher (Williams College) and Andrew Granville (University of Mon
treal)\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/108/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Myrto Mavraki (Harvard University)
DTSTART:20211118T160000Z
DTEND:20211118T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/109
DESCRIPTION:Title: Towards uniformity in the dynamical Bogomolov conjecture
\nby Myrto Mavraki (Harvard University) as part of Number Theory Web Semin
ar\n\n\nAbstract\nInspired by an analogy between torsion and preperiodic p
oints\, Zhang has proposed a dynamical generalization of the classical Man
in-Mumford and Bogomolov conjectures. A special case of these conjectures\
, for `split' maps\, has recently been established by Nguyen\, Ghioca and
Ye. In particular\, they show that two rational maps have at most finitely
many common preperiodic points\, unless they are `related'. Recent breakt
hroughs by Dimitrov\, Gao\, Habegger and Kühne have established that the
classical Bogomolov conjecture holds uniformly across curves of given genu
s. \n\nIn this talk we discuss uniform versions of the dynamical Bogomolov
conjecture across 1-parameter families of certain split maps. To this end
\, we establish an instance of a 'relative dynamical Bogomolov'. This is w
ork in progress joint with Harry Schmidt (University of Basel).\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/109/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Johan Commelin (Albert–Ludwigs-Universität Freiburg)
DTSTART:20211021T150000Z
DTEND:20211021T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/110
DESCRIPTION:Title: Liquid Tensor Experiment\nby Johan Commelin (Albert–Lu
dwigs-Universität Freiburg) as part of Number Theory Web Seminar\n\n\nAbs
tract\nIn December 2020\, Peter Scholze posed a challenge to formally veri
fy the main theorem on liquid $\\mathbb{R}$-vector spaces\, which is part
of his joint work with Dustin Clausen on condensed mathematics. I took up
this challenge with a team of mathematicians to verify the theorem in the
Lean proof assistant. Half a year later\, we reached a major milestone\, a
nd our expectation is that in a couple of months we will have completed th
e full challenge.\n\nIn this talk I will give a brief motivation for conde
nsed/liquid mathematics\, a demonstration of the Lean proof assistant\, an
d discuss our experiences formalizing state-of-the-art research in mathema
tics.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/110/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zeev Dvir (Princeton University)
DTSTART:20210819T150000Z
DTEND:20210819T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/111
DESCRIPTION:Title: The Kakeya set conjecture over rings of integers modulo squa
re free $m$\nby Zeev Dvir (Princeton University) as part of Number The
ory Web Seminar\n\n\nAbstract\nWe show that\, when $N$ is any square-free
integer\, the size of the smallest Kakeya set in $(ℤ/Nℤ)^n$ is at leas
t $C_{\\epsilon\,n}N^{n-\\epsilon}$ for any $\\epsilon>0$ -- resolving a s
pecial case of a conjecture of Hickman and Wright. Previously\, such bound
s were only known for the case of prime $N$. We also show that the case of
general $N$ can be reduced to lower bounding the $p$-rank of the incidenc
e matrix of points and hyperplanes over $(ℤ/p^kℤ)^n$. Joint work with
Manik Dhar.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/111/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexei Skorobogatov (Imperial College London)
DTSTART:20211125T160000Z
DTEND:20211125T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/112
DESCRIPTION:Title: On uniformity conjectures for abelian varieties and K3 surfa
ces\nby Alexei Skorobogatov (Imperial College London) as part of Numbe
r Theory Web Seminar\n\n\nAbstract\nI will discuss logical links among uni
formity conjectures concerning K3 surfaces and abelian varieties of bounde
d dimension defined over number fields of bounded degree. The conjectures
concern the endomorphism algebra of an abelian variety\, the Néron–Seve
ri lattice of a K3 surface\, and the Galois invariant subgroup of the geom
etric Brauer group. The talk is based on a joint work with Martin Orr and
Yuri Zarhin.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/112/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katherine Stange (University of Colorado\, Boulder)
DTSTART:20211104T160000Z
DTEND:20211104T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/113
DESCRIPTION:Title: Algebraic Number Starscapes\nby Katherine Stange (Univer
sity of Colorado\, Boulder) as part of Number Theory Web Seminar\n\n\nAbst
ract\nIn the spirit of experimentation\, at the Fall 2019 ICERM special se
mester on “Illustrating Mathematics\,” I began drawing algebraic numbe
rs in the complex plane. Edmund Harriss\, Steve Trettel and I sized the n
umbers by arithmetic complexity and found a wealth of pattern and structur
e. In this talk\, I’ll take you on a visual tour and share some of the
mathematical explanations we found for what can be quite stunning pictures
(in the hands of a mathematician and artist like Edmund). This experienc
e gave me a new perspective on complex Diophantine approximation: one can
view approximation properties as being dictated by the geometry of the ma
p from coefficient space to root space in different polynomial degrees. I
’ll explain this geometry\, and discuss a few Diophantine results\, know
n and new\, in this context.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/113/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kiran Kedlaya (University of California San Diego)
DTSTART:20211202T160000Z
DTEND:20211202T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/114
DESCRIPTION:Title: Orders of abelian varieties over $\\mathbb{F}_2$\nby Kir
an Kedlaya (University of California San Diego) as part of Number Theory W
eb Seminar\n\n\nAbstract\nWe describe several recent results on orders of
abelian varieties over $\\mathbb{F}_2$: every positive integer occurs as t
he order of an ordinary abelian variety over $\\mathbb{F}_2$ (joint with E
. Howe)\; every positive integer occurs infinitely often as the order of a
simple abelian variety over $\\mathbb{F}_2$\; the geometric decomposition
of the simple abelian varieties over $\\mathbb{F}_2$ can be described exp
licitly (joint with T. D'Nelly-Warady)\; and the relative class number one
problem for function fields is reduced to a finite computation (work in p
rogress). All of these results rely on the relationship between isogeny cl
asses of abelian varieties over finite fields and Weil polynomials given b
y the work of Weil and Honda-Tate. With these results in hand\, most of th
e work is to construct algebraic integers satisfying suitable archimedean
constraints.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/114/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Avi Wigderson (Institute for Advanced Study)
DTSTART:20211111T160000Z
DTEND:20211111T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/115
DESCRIPTION:Title: Randomness\nby Avi Wigderson (Institute for Advanced Stu
dy) as part of Number Theory Web Seminar\n\n\nAbstract\nIs the universe in
herently deterministic or probabilistic? Perhaps more importantly - can we
tell the difference between the two? \n\nHumanity has pondered the meanin
g and utility of randomness for millennia. \nThere is a remarkable variety
of ways in which we utilize perfect coin tosses to our advantage: in stat
istics\, cryptography\, game theory\, algorithms\, gambling... Indeed\, ra
ndomness seems indispensable! Which of these applications survive if the u
niverse had no (accessible) randomness in it at all? Which of them survive
if only poor quality randomness is available\, e.g. that arises from some
what "unpredictable" phenomena like the weather or the stock market? \n\nA
computational theory of randomness\, developed in the past several decade
s\, reveals (perhaps counter-intuitively) that very little is lost in such
deterministic or weakly random worlds. In the talk I'll explain the main
ideas and results of this theory\, notions of pseudo-randomness\, and conn
ections to computational intractability. \n\nIt is interesting that Number
Theory played an important role throughout this development. It supplied
problems whose algorithmic solution make randomness seem powerful\, proble
ms for which randomness can be eliminated from such solutions\, and proble
ms where the power of randomness remains a major challenge for computation
al complexity theorists and mathematicians. I will use these problems (and
others) to demonstrate aspects of this theory.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/115/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sarah Zerbes (University College London\, UK)
DTSTART:20211216T160000Z
DTEND:20211216T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/116
DESCRIPTION:Title: Euler systems and the Birch—Swinnerton-Dyer conjecture for
abelian surfaces\nby Sarah Zerbes (University College London\, UK) as
part of Number Theory Web Seminar\n\n\nAbstract\nEuler systems are one of
the most powerful tools for proving cases of the Bloch--Kato conjecture\,
and other related problems such as the Birch and Swinnerton-Dyer conjectu
re. \n\nI will recall a series of recent works (variously joint with Loeff
ler\, Pilloni\, Skinner) giving rise to an Euler system in the cohomology
of Shimura varieties for $\\mathrm{GSp}(4)$\, and an explicit reciprocity
law relating the Euler system to values of $L$-functions. I will then rece
nt work with Loeffler\, in which we use this Euler system to prove new cas
es of the BSD conjecture for modular abelian surfaces over $\\Q$\, and mod
ular elliptic curves over imaginary quadratic fields.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/116/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samir Siksek (University of Warwick)
DTSTART:20211209T160000Z
DTEND:20211209T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/117
DESCRIPTION:Title: The Fermat equation and the unit equation\nby Samir Siks
ek (University of Warwick) as part of Number Theory Web Seminar\n\n\nAbstr
act\nThe asymptotic Fermat conjecture (AFC) states that for a number field
$K$\, and for sufficiently large primes $p$\, the only solutions to the F
ermat equation $X^p+Y^p+Z^p=0$ in $K$ are the obvious ones. We sketch rece
nt work that connects the Fermat equation to the far more elementary unit
equation\, and explain how this surprising connection can be exploited to
prove AFC for several infinite families of number fields. This talk is bas
ed on joint work with Nuno Freitas\, Alain Kraus and Haluk Sengun.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/117/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Péter Varjú (University of Cambridge)
DTSTART:20220113T160000Z
DTEND:20220113T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/118
DESCRIPTION:Title: Irreducibility of random polynomials\nby Péter Varjú (
University of Cambridge) as part of Number Theory Web Seminar\n\n\nAbstrac
t\nConsider random polynomials of degree $d$ whose leading and constant co
efficients are $1$ and the rest are independent taking the values $0$ or $
1$ with equal probability. A conjecture of Odlyzko and Poonen predicts th
at such a polynomial is irreducible in $\\Z[x]$ with high probability as $
d$ grows. This conjecture is still open\, but Emmanuel Breuillard and I pr
oved it assuming the Extended Riemann Hypothesis. I will briefly recall th
e method of proof of this result and will discuss later developments that
apply this method to other models of random polynomials.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/118/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ekin Özman (Boğaziçi University)
DTSTART:20220303T160000Z
DTEND:20220303T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/119
DESCRIPTION:Title: Modular Curves and Asymptotic Solutions to Fermat-type Equat
ions\nby Ekin Özman (Boğaziçi University) as part of Number Theory
Web Seminar\n\n\nAbstract\nUnderstanding solutions of Diophantine equation
s over rationals or more generally over any number field is one of the mai
n problems of number theory. By the help of the modular techniques used in
the proof of Fermat’s last theorem by Wiles and its generalizations\, i
t is possible to solve other Diophantine equations too. Understanding quad
ratic points on the classical modular curve play a central role in this ap
proach. It is also possible to study the solutions of Fermat type equation
s over number fields asymptotically. In this talk\, I will mention some re
cent results about these notions for the classical Fermat equation as well
as some other Diophantine equations.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/119/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ana Caraiani (Imperial College London)
DTSTART:20220407T150000Z
DTEND:20220407T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/120
DESCRIPTION:Title: On the cohomology of Shimura varieties with torsion coeffici
ents\nby Ana Caraiani (Imperial College London) as part of Number Theo
ry Web Seminar\n\n\nAbstract\nShimura varieties are certain highly symmetr
ic algebraic varieties that generalise modular curves and that play an imp
ortant role in the Langlands program. In this talk\, I will survey recent
vanishing conjectures and results about the cohomology of Shimura varietie
s with torsion coefficients\, under both local and global representation-t
heoretic conditions. I will illustrate the geometric ingredients needed to
establish these results using the toy model of the modular curve. I will
also mention several applications\, including to (potential) modularity ov
er CM fields.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/120/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Larry Guth (MIT)
DTSTART:20220127T160000Z
DTEND:20220127T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/121
DESCRIPTION:Title: Reflections on the proof(s) of the Vinogradov mean value con
jecture\nby Larry Guth (MIT) as part of Number Theory Web Seminar\n\n\
nAbstract\nThe Vinogradov mean value conjecture concerns the number of sol
utions of a system of diophantine equations. This number of solutions can
also be written as a certain moment of a trigonometric polynomial. The c
onjecture was proven in the 2010s by Bourgain-Demeter-Guth and by Wooley\,
and recently there was a shorter proof by Guo-Li-Yang-Zorin-Kranich. The
details of each proof involve some intricate estimates. The goal of the t
alk is to try to reflect on the proof(s) in a big picture way. A key ingr
edient in all the proofs is to combine estimates at many different scales\
, usually by doing induction on scales. Why does this multi-scale inducti
on help? What can multi-scale induction tell us and what are its limitati
ons?\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/121/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Humphries (University of Virginia)
DTSTART:20220203T160000Z
DTEND:20220203T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/122
DESCRIPTION:Title: $L^p$-norm bounds for automorphic forms\nby Peter Humphr
ies (University of Virginia) as part of Number Theory Web Seminar\n\n\nAbs
tract\nA major area of study in analysis involves the distribution of mass
of Laplacian eigenfunctions on a Riemannian manifold. A key result toward
s this is explicit $L^p$-norm bounds for Laplacian eigenfunctions in terms
of their Laplacian eigenvalue\, due to Sogge in 1988. Sogge's bounds are
sharp on the sphere\, but need not be sharp on other manifolds. I will dis
cuss some aspects of this problem for the modular surface\; in this settin
g\, the Laplacian eigenfunctions are automorphic forms\, and certain $L^p$
-norms can be shown to be closely related to certain mixed moments of $L$-
functions. This is joint with with Rizwanur Khan.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/122/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ram Murty (Queen's University)
DTSTART:20220414T150000Z
DTEND:20220414T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/124
DESCRIPTION:Title: Probability Theory and the Riemann Hypothesis\nby Ram Mu
rty (Queen's University) as part of Number Theory Web Seminar\n\n\nAbstrac
t\nThere is a probability distribution attached to the Riemann zeta functi
on which allows one to formulate the Riemann hypothesis in terms of the cu
mulants of this distribution and is due to Biane\, Pitman and Yor. The cum
ulants can be related to generalized Euler-Stieltjes constants and to Li's
criterion for the Riemann hypothesis. We will discuss these results and
present some new results related to this theme.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/124/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jozsef Solymosi (University of British Columbia)
DTSTART:20220120T160000Z
DTEND:20220120T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/125
DESCRIPTION:Title: Rank of matrices with entries from a multiplicative group\nby Jozsef Solymosi (University of British Columbia) as part of Number T
heory Web Seminar\n\n\nAbstract\nWe establish lower bounds on the rank of
matrices in which all but the diagonal entries lie in a multiplicative gro
up of small rank. Applying these bounds we show that the distance sets of
finite pointsets in $\\R^d$ generate high rank multiplicative groups and t
hat multiplicative groups of small rank cannot contain large sumsets. (Joi
nt work with Noga Alon)\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/125/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dmitry Kleinbock (Brandeis University)
DTSTART:20220310T160000Z
DTEND:20220310T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/126
DESCRIPTION:Title: Shrinking targets on homogeneous spaces and improving Dirich
let's Theorem\nby Dmitry Kleinbock (Brandeis University) as part of Nu
mber Theory Web Seminar\n\n\nAbstract\nLet $\\psi$ be a decreasing functio
n defined on all large positive real numbers. We say that a real $m \\time
s n$ matrix $Y$ is "$\\psi$-Dirichlet" if for every sufficiently large rea
l number $T$ there exist non-trivial integer vectors $(p\,q)$ satisfying $
\\|Yq-p\\|^m < \\psi(T)$ and $\\|q\\|^n < T$ (where $\\|\\cdot\\|$ denotes
the supremum norm on vectors). This generalizes the property of $Y$ being
"Dirichlet improvable" which has been studied by several people\, startin
g with Davenport and Schmidt in 1969. I will present results giving suffic
ient conditions on $\\psi$ to ensure that the set of $\\psi$-Dirichlet mat
rices has zero (resp.\, full) measure. If time allows I will mention a geo
metric generalization of the set-up\, where the supremum norm is replaced
by an arbitrary norm. Joint work with Anurag Rao\, Andreas Strombergsson\,
Nick Wadleigh and Shuchweng Yu.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/126/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Levin (Michigan State University)
DTSTART:20220317T160000Z
DTEND:20220317T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/127
DESCRIPTION:Title: Diophantine Approximation for Closed Subschemes\nby Aaro
n Levin (Michigan State University) as part of Number Theory Web Seminar\n
\n\nAbstract\nThe classical Weil height machine associates heights to divi
sors on a projective variety. I will give a brief\, but gentle\, introduct
ion to how this machinery extends to objects (closed subschemes) in higher
codimension\, due to Silverman\, and discuss various ways to interpret th
e heights. We will then discuss several recent results in which these idea
s play a prominent and central role.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/127/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Granville (Université de Montréal)
DTSTART:20220428T150000Z
DTEND:20220428T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/128
DESCRIPTION:Title: Linear Divisibility sequences\nby Andrew Granville (Univ
ersité de Montréal) as part of Number Theory Web Seminar\n\n\nAbstract\n
In 1878\, in the first volume of the first mathematics journal published i
n the US\, Edouard Lucas wrote 88 pages (in French) on linear recurrence s
equences\, placing Fibonacci numbers and other linear recurrence sequences
into a broader context. He examined their behaviour locally as well as gl
obally\, and asked several questions that influenced much research in the
century and a half to come.\n\nIn a sequence of papers in the 1930s\, Mars
hall Hall further developed several of Lucas' themes\, including studying
and trying to classify third order linear divisibility sequences\; that is
\, linear recurrences like the Fibonacci numbers which have the additional
property that $F_m$ divides $F_n$ whenever $m$ divides $n$. Because of ma
ny special cases\, Hall was unable to even conjecture what a general theo
rem should look like\, and despite developments over the years by various
authors\, such as Lehmer\, Morgan Ward\, van der Poorten\, Bezivin\, Petho
\, Richard Guy\, Hugh Williams\,... with higher order linear divisibility
sequences\, even the formulation of the classification has remained myster
ious.\n\nIn this talk we present our ongoing efforts to classify all linea
r divisibility sequences\, the key new input coming from a wonderful appli
cation of the Schmidt/Schlickewei subspace theorem from the theory of diop
hantine approximation\, due to Corvaja and Zannier.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/128/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Harry Schmidt (University of Basel)
DTSTART:20220217T160000Z
DTEND:20220217T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/129
DESCRIPTION:Title: Counting rational points and lower bounds for Galois orbits
for special points on Shimura varieties\nby Harry Schmidt (University
of Basel) as part of Number Theory Web Seminar\n\n\nAbstract\nIn this talk
I will give an overview of the history of the André-Oort conjecture and
its resolution last year after the final steps were made in work of Pila\,
Shankar\, Tsimerman\, Esnault and Groechenig as well as Binyamini\, Yafae
v and myself. I will focus on the key insights and ideas related to model
theory and transcendence theory.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/129/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Levent Alpöge (Harvard University)
DTSTART:20220505T150000Z
DTEND:20220505T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/132
DESCRIPTION:Title: On integers which are(n't) the sum of two rational cubes
\nby Levent Alpöge (Harvard University) as part of Number Theory Web Semi
nar\n\n\nAbstract\nIt's easy that $0\\%$ of integers are the sum of two in
tegral cubes (allowing opposite signs!).\nI will explain joint work with B
hargava and Shnidman in which we show:\n\n1. At least a sixth of integers
are not the sum of two rational cubes\,\n\nand\n\n2. At least a sixth of o
dd integers are the sum of two rational cubes!\n(--- with 2. relying on ne
w $2$-converse results of Burungale-Skinner.)\n\nThe basic principle is th
at "there aren't even enough $2$-Selmer elements to go around" to contradi
ct e.g. 1.\, and we show this by using the circle method "inside" the usua
l geometry of numbers argument applied to a particular coregular represent
ation. Even then the resulting constant isn't small enough to conclude 1.\
, so we use the clean form of root numbers in the family $x^3 + y^3 = n$ a
nd the $p$-parity theorem of Nekovar/Dokchitser-Dokchitser to succeed.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/132/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Winnie Li (Pennsylvania State University)
DTSTART:20220324T160000Z
DTEND:20220324T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/133
DESCRIPTION:Title: Group based zeta functions\nby Winnie Li (Pennsylvania S
tate University) as part of Number Theory Web Seminar\n\n\nAbstract\nThe t
heme of this survey talk is zeta functions which count closed geodesics on
objects arising from real and $p$-adic groups. Our focus is on $\\PGL(n)$
. For $\\PGL(2)$\, these are the Selberg zeta function for compact quotien
ts of the upper half-plane and the Ihara zeta function for finite regular
graphs. We shall explain the identities satisfied by these zeta functions\
, which show interconnections between combinatorics\, group theory and num
ber theory. Comparisons will be made for zeta identities from different ba
ckground. Like the Riemann zeta function\, the analytic behavior of a gro
up based zeta function governs the distribution of the prime geodesics in
its definition.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/133/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joni Teräväinen (University of Turku)
DTSTART:20220421T150000Z
DTEND:20220421T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/134
DESCRIPTION:Title: Short exponential sums of the primes\nby Joni Teräväin
en (University of Turku) as part of Number Theory Web Seminar\n\n\nAbstrac
t\nI will discuss the short interval behaviour of the von Mangoldt and Mö
bius functions twisted by exponentials. I will in particular mention new r
esults on sums of these functions twisted by polynomial exponential phases
\, or even more general nilsequence phases. I will also discuss connection
s to Chowla's conjecture. This is based on joint works with Kaisa Matomäk
i\, Maksym Radziwiłł\, Xuancheng Shao\, Terence Tao and Tamar Ziegler.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/134/
END:VEVENT
BEGIN:VEVENT
SUMMARY:William Chen (Institute for Advanced Study)
DTSTART:20220331T150000Z
DTEND:20220331T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/135
DESCRIPTION:Title: Markoff triples and connectivity of Hurwitz spaces\nby W
illiam Chen (Institute for Advanced Study) as part of Number Theory Web Se
minar\n\n\nAbstract\nIn this talk we will show that the integral points of
the Markoff equation $x^2 + y^2 + z^2 - xyz = 0$ surject onto its $F_p$-p
oints for all but finitely many primes $p$. This essentially resolves a co
njecture of Bourgain\, Gamburd\, and Sarnak\, and a question of Frobenius
from 1913. The proof relates the question to the classical problem of clas
sifying the connected components of the Hurwitz moduli spaces $H(g\,n)$ cl
assifying finite covers of genus $g$ curves with $n$ branch points. Over a
century ago\, Clebsch and Hurwitz established connectivity for the subspa
ce classifying simply branched covers of the projective line\, which led t
o the first proof of the irreducibility of the moduli space of curves of a
given genus. More recently\, the work of Dunfield-Thurston and Conway-Par
ker establish connectivity in certain situations where the monodromy group
is fixed and either $g$ or $n$ are allowed to be large\, which has been a
pplied to study Cohen-Lenstra heuristics over function fields. In the case
where $(g\,n)$ are fixed and the monodromy group is allowed to vary\, far
less is known. In our case we study $\\SL(2\,p)$-covers of elliptic curve
s\, only branched over the origin\, and establish connectivity\, for all s
ufficiently large p\, of the subspace classifying those covers with ramifi
cation indices $2p$. The proof builds upon asymptotic results of Bourgain\
, Gamburd\, and Sarnak\, the key new ingredient being a divisibility resul
t on the degree of a certain forgetful map between moduli spaces\, which p
rovides enough rigidity to bootstrap their asymptotics to a result for all
sufficiently large $p$.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/135/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elon Lindenstrauss (Hebrew University of Jerusalem)
DTSTART:20220602T150000Z
DTEND:20220602T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/136
DESCRIPTION:Title: Effective equidistribution of some unipotent flows with poly
nomial rates\nby Elon Lindenstrauss (Hebrew University of Jerusalem) a
s part of Number Theory Web Seminar\n\n\nAbstract\nJoint work with Amir Mo
hammadi and Zhiren Wang\n\nA landmark result of Ratner gives that if $G$ i
s a real linear algebraic group\, $\\Gamma$ a lattice in $G$ and if $u_t$
is a one-parameter unipotent subgroup of $G$\, then for any $x \\in G/\\Ga
mma$ the orbit $u_t.x$ is equidistributed in a periodic orbit of some subg
roup $L < G$\, and moreover that the orbit of $x$ under $u_t$ is contained
in this periodic $L$ orbit.\n\nA key motivation behind Ratner's equidistr
ibution theorem for one-parameter unipotent flows has been to establish Ra
ghunathan's conjecture regarding the possible orbit closures of groups gen
erated by one-parameter unipotent groups\; using the equidistribution theo
rem Ratner proved that if $G$ and $\\Gamma$ are as above\, and if $H < G$
is generated by one parameter unipotent groups then for any $x \\in G/\\Ga
mma$ one has that $\\overline{H.x}=L.x$ where $H < L < G$ and $L.x$ is per
iodic. Important special cases of Raghunathan's conjecture were proven ear
lier by Margulis and by Dani and Margulis by a different\, more direct\, a
pproach.\n\nThese results have had many beautiful and unexpected applicati
ons in number theory\, geometry and other areas. A key challenge has been
to quantify and effectify these results. Beyond the case of actions of hor
ospheric groups where there are several fully quantitative and effective r
esults available\, results in this direction have been few and far between
. In particular\, if $G$ is semisimple and $U$ is not horospheric no quant
itative form of Ratner's equidistribution was known with any error rate\,
though there has been some progress on understanding quantitatively densit
y properties of such flows with iterative logarithm error rates.\n\nIn my
talk I will present a new fully quantitative and effective equidistributio
n result for orbits of one-parameter unipotent groups in arithmetic quotie
nts of $\\SL_2(\\C)$ and $\\SL_2(\\R)\\times\\SL(2\,\\R)$. I will also try
to explain a bit the connection to number theory.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/136/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yunqing Tang (Princeton University)
DTSTART:20220526T150000Z
DTEND:20220526T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/137
DESCRIPTION:Title: Applications of arithmetic holonomicity theorems\nby Yun
qing Tang (Princeton University) as part of Number Theory Web Seminar\n\n\
nAbstract\nIn this talk\, we will discuss the proof of the unbounded denom
inators conjecture on Fourier coefficients of $\\SL_2(\\Z)$-modular forms\
, and the proof of irrationality of $2$-adic zeta value at $5$. Both proof
s use an arithmetic holonomicity theorem\, which can be viewed as a refine
ment of André’s algebraicity criterion. If time permits\, we will give
a proof of the arithmetic holonomicity theorem via the slope method a la B
ost.\nThis is joint work with Frank Calegari and Vesselin Dimitrov.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/137/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zeev Rudnick (Tel Aviv University)
DTSTART:20220210T160000Z
DTEND:20220210T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/138
DESCRIPTION:Title: Beyond uniform distribution\nby Zeev Rudnick (Tel Aviv U
niversity) as part of Number Theory Web Seminar\n\n\nAbstract\nThe study o
f uniform distribution of sequences is more than a century old\, with pion
eering work by Hardy and Littlewood\, Weyl\, van der Corput and others. Mo
re recently\, the focus of research has shifted to much finer quantities\,
such as the distribution of nearest neighbor gaps and the pair correlatio
n function. Examples of interesting sequences for which these quantities h
ave been studied include the zeros of the Riemann zeta function\, energy l
evels of quantum systems\, and more. In this expository talk\, I will disc
uss what is known about these examples and discuss the many outstanding pr
oblems that this theory has to offer.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/138/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicole Looper (Brown University)
DTSTART:20220609T150000Z
DTEND:20220609T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/139
DESCRIPTION:Title: The Uniform Boundedness Principle for polynomials over numbe
r fields\nby Nicole Looper (Brown University) as part of Number Theory
Web Seminar\n\n\nAbstract\nThis talk is about uniform bounds on the numbe
r of $K$-rational preperiodic points across families of endomorphisms of p
rojective space defined over various fields $K$. We will focus on the case
where $K$ is a number field\, and the morphisms are polynomial maps on $\
\mathbb{P}^1$. Along the way\, I will highlight the more challenging aspec
ts behind the known approaches\, and discuss the obstacles to be addressed
in future research.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/139/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amir Shpilka (Tel Aviv University)
DTSTART:20220623T150000Z
DTEND:20220623T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/140
DESCRIPTION:Title: Points\, lines and polynomial identities\nby Amir Shpilk
a (Tel Aviv University) as part of Number Theory Web Seminar\n\n\nAbstract
\nThe Sylvester-Gallai (SG) theorem in discrete geometry asserts that if a
finite set of points P has the property that every line through any two o
f its points intersects the set at a third point\, then P must lie on a li
ne. Surprisingly\, this theorem\, and some variants of it\, appear in the
analysis of locally correctable codes and\, more noticeably\, in algebraic
program testing (polynomial identity testing). For these questions one of
ten has to study extensions of the original SG problem: the case where the
re are several sets\, or with a robust version of the condition (many "spe
cial" lines through each point) or with a higher degree analog of the prob
lem\, etc.\n\nIn this talk I will present the SG theorem and some of its v
ariants\, show its relation to the above mentioned computational problems
and discuss recent developments regarding higher degree analogs and their
applications.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/140/
END:VEVENT
BEGIN:VEVENT
SUMMARY:John Voight (Dartmouth College)
DTSTART:20220616T150000Z
DTEND:20220616T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/141
DESCRIPTION:Title: Counting elliptic curves with level structure\nby John V
oight (Dartmouth College) as part of Number Theory Web Seminar\n\n\nAbstra
ct\nFollowing work of Harron and Snowden\, we provide an asymptotic answer
to questions like: how many elliptic curves of bounded height have a cycl
ic isogeny of degree $N$? We'll begin\nwith a survey the recent spate of w
ork on this topic\, and then we will report on joint work with Carl Pomera
nce and Maggie Pizzo\, with John Cullinan and Meagan Kenney\, and finally\
nwith Grant Molnar.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/141/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Shparlinski (UNSW Sydney)
DTSTART:20220224T160000Z
DTEND:20220224T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/142
DESCRIPTION:Title: Sums of Kloosterman and Salie Sums and Moments of $L$-funct
ions\nby Igor Shparlinski (UNSW Sydney) as part of Number Theory Web S
eminar\n\n\nAbstract\nWe present some old and more recent results which su
ggest that Kloosterman and Salie sums exhibit a pseudorandom behaviour sim
ilar to the behaviour which is traditionally attributed to the Mobius func
tion. In particular\, we formulate some analogues of the Chowla Conjecture
for Kloosterman and Salie sums. We then describe several results about th
e non-correlation of Kloosterman and Salie sums between themselves and als
o with some classical number-theoretic functions such as the Mobius functi
on\, the divisor function and the sums of binary digits. Various arithmeti
c applications of these results\, including to asymptotic formulas for mom
ents of various $L$-functions\, will be outlined as well.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/142/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Gamburd (CUNY Graduate Center)
DTSTART:20220630T150000Z
DTEND:20220630T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/143
DESCRIPTION:Title: Arithmetic and dynamics on varieties of Markoff type\nby
Alexander Gamburd (CUNY Graduate Center) as part of Number Theory Web Sem
inar\n\n\nAbstract\nThe Markoff equation $x^2+y^2+z^2=3xyz$\, which arose
in his spectacular thesis (1879)\, is ubiquitous in a tremendous variety o
f contexts. After reviewing some of these\, we will discuss recent progres
s towards establishing forms of strong approximation on varieties of Marko
ff type\, as well as ensuing implications\, diophantine and dynamical.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/143/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Vaaler (University of Texas at Austin)
DTSTART:20220519T150000Z
DTEND:20220519T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/144
DESCRIPTION:Title: Schinzel's determinant inequality and a conjecture of F. Rod
riguez Villegas\nby Jeffrey Vaaler (University of Texas at Austin) as
part of Number Theory Web Seminar\n\n\nAbstract\nThe Abstract is available
at\n\nhttps://www.ntwebseminar.org/home\n\nor directly at\n\nhttps://driv
e.google.com/file/d/1VDQLDlcC3IDEMduR6H-X9Rf0jRxSZ_J-/view\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/144/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Charles Vaughan (Pennsylvania State University)
DTSTART:20220512T150000Z
DTEND:20220512T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/145
DESCRIPTION:Title: Generalizations of the Montgomery-Hooley asymptotic formula\
; A survey.\nby Robert Charles Vaughan (Pennsylvania State University)
as part of Number Theory Web Seminar\n\n\nAbstract\nFollowing a statement
without proof in a special case by Barban [1966]\, and less precise bound
s by Davenport and Halberstam [1966] and Gallagher [1967]\, Montgomery [19
70] obtained the asymptotic formula\n\\[\n\\sum_{q\\le Q} \\sum_{\\stackre
l{a=1}{(a\,q)=1}}^q \\left|\n\\psi(x\;q\,a) - \\frac{x}{\\phi(q)}\n\\right
|^2 \\sim Qx\\log x\n\\]\nvalid when $x(\\log x)^{-A}\\le Q\\le x$ and $A$
is fixed. This was refined and the proof substantially simplified by Hoo
ley [1975] in the first of a celebrated series of 19 papers with the gener
ic title ``On the Barban-Davenport-Halberstam theorem" which have widened
the scope of the methods. There have been also a number of papers by othe
r authors with further generalizations and I will give a survey of this to
gether with an overview of some of the recent developments.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/145/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandra Florea (University of California Irvine)
DTSTART:20220922T150000Z
DTEND:20220922T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/146
DESCRIPTION:Title: Negative moments of the Riemann zeta function\nby Alexan
dra Florea (University of California Irvine) as part of Number Theory Web
Seminar\n\n\nAbstract\nI will talk about work towards a conjecture of Gone
k regarding negative shifted moments of the Riemann zeta function. I will
explain how to obtain asymptotic formulas when the shift in the Riemann ze
ta function is big enough\, and how one can obtain non-trivial upper bound
s for smaller shifts. Joint work with H. Bui.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/146/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ping Xi (Xi'an Jiaotong University)
DTSTART:20220908T150000Z
DTEND:20220908T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/147
DESCRIPTION:Title: Analytic approaches towards Katz’s problems on Kloosterman
sums\nby Ping Xi (Xi'an Jiaotong University) as part of Number Theory
Web Seminar\n\n\nAbstract\nMotivated by deep observations on elliptic cur
ves/modular forms\, Nicholas Katz proposed three problems on sign changes\
, equidistributions and modular structures of Kloosterman sums in 1980. In
this talk\, we will discuss some recent progresses towards these three pr
oblems made by analytic number theory (e.g.\, sieve methods and automorphi
c forms) combining certain tools from $\\ell$-adic cohomology.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/147/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yann Bugeaud (University of Strasbourg)
DTSTART:20220901T150000Z
DTEND:20220901T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/148
DESCRIPTION:Title: $B'$\nby Yann Bugeaud (University of Strasbourg) as part
of Number Theory Web Seminar\n\n\nAbstract\nLet $n \\ge 1$ be an integer
and $\\alpha_1\, \\ldots\, \\alpha_n$ be non-zero algebraic numbers. \nLet
$b_1\, \\ldots \, b_n$ be integers with $b_n \\not= 0$\, and set $B = \\m
ax\\{3\, |b_1|\, \\ldots \, |b_n|\\}$. \nFor $j =1\, \\ldots\, n$\, set $h
^* (\\alpha_j) = \\max\\{h(\\alpha_j)\, 2\\}$\, where $h$ \ndenotes the (l
ogarithmic) Weil height. \nAssume that the quantity $\\Lambda = b_1 \\log
\\alpha_1 + \\cdots + b_n \\log \\alpha_n$ is nonzero. \nA typical lower b
ound of $\\log |\\Lambda|$ given by Baker's theory of linear forms in loga
rithms takes the shape \n$$\n- c(n\, D) \\\, h^* (\\alpha_1) \\ldots h^*
(\\alpha_n) \\log B\, \n$$\nwhere $c(n\,D)$ is positive\, effectively comp
utable and depends only on $n$ and on the degree $D$ of the field generate
d \nby $\\alpha_1\, \\ldots \, \\alpha_n$. \nHowever\, in certain special
cases and in particular when $|b_n| = 1$\, this bound can be improved to\n
$$\n- c(n\, D) \\\, h^* (\\alpha_1) \\ldots h^*(\\alpha_n) \\log \\frac
{B}{h^*(\\alpha_n)}.\n$$\nThe term $B' := B / h^*(\\alpha_n)$ in place of
$B$ \noriginates in works of Feldman and of Baker. It is a key tool for im
proving\, in an effective way\, the upper bound for the irrationality expo
nent\nof a real algebraic number of degree at least $3$ given by \nLiouvil
le's theorem.\nWe survey various applications of this $B'$ to exponents of
approximation evaluated at algebraic numbers\, \nto the $S$-part of integ
er sequences\, and to Diophantine equations.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/148/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paul Nelson (Aarhus University)
DTSTART:20220929T150000Z
DTEND:20220929T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/149
DESCRIPTION:Title: The sup norm problem in the level aspect\nby Paul Nelson
(Aarhus University) as part of Number Theory Web Seminar\n\n\nAbstract\nT
he sup norm problem concerns the size of $L^2$-normalized eigenfunctions o
f manifolds. In many situations\, one expects to be able to improve upon
the general bound following from local considerations. The pioneering res
ult in that direction is due to Iwaniec and Sarnak\, who in 1995 establish
ed an improvement upon the local bound for Hecke-Maass forms of large eige
nvalue on the modular surface. Their method has since been extended and a
pplied by many authors\, notably to the "level aspect" variant of the prob
lem\, where one varies the underlying manifold rather than the eigenvalue.
Recently\, Raphael Steiner introduced a new method for attacking the sup
norm problem. I will describe joint work with Raphael Steiner and Ilya K
hayutin in which we apply that method to improve upon the best known bound
s in the level aspect.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/149/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey C. Lagarias (University of Michigan)
DTSTART:20221006T150000Z
DTEND:20221006T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/150
DESCRIPTION:Title: The Alternative Hypothesis and Point Processes\nby Jeffr
ey C. Lagarias (University of Michigan) as part of Number Theory Web Semin
ar\n\n\nAbstract\nThe Alternative Hypothesis concerns a hypothetical and u
nlikely picture of how zeros of the Riemann zeta function are spaced. It a
sks that nearly all normalized zero spacings be near half-integers. Thi
s possible zero distribution is incompatible with the GUE distribution of
zero spacings. Ruling it out arose as an obstacle to the long-standing pr
oblem of proving there are no exceptional zeros of Dirichlet $L$-function
s. The talk describes joint work with Brad Rodgers\, that constructs a poi
nt process realizing Alternative Hypothesis type statistics\, which is c
onsistent with the known results on correlation functions for spacings of
zeta zeros. (A similar result was independently obtained by Tao with sli
ghtly different methods.) The talk reviews point process models and prese
nts further results on the general problem of to what extent two point pro
cesses\, a continuous one on the real line\, the other a discrete one on a
lattice $a\\Z$\, can mimic each other in the sense of having perfect agr
eement of all their correlation functions when convolved with bandlimited
test functions of a given bandwidth $B$.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/150/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shai Evra (Hebrew University of Jerusalem)
DTSTART:20221103T160000Z
DTEND:20221103T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/151
DESCRIPTION:Title: Optimal strong approximation and the Sarnak-Xue density hypo
thesis\nby Shai Evra (Hebrew University of Jerusalem) as part of Numbe
r Theory Web Seminar\n\n\nAbstract\nIt is a classical result that the modu
lo map from $\\SL_2(\\Z)$ to $\\SL_2(\\Z/q\\Z)$\, is surjective for any in
teger $q$. The generalization of this phenomenon to other arithmetic group
s goes under the name of strong approximation\, and it is well understood.
The following natural question was recently raised in a letter of Sarnak:
What is the minimal exponent $e$\, such that for any large $q$\, almost a
ny element of $\\SL_2(\\Z/q\\Z)$ has a lift in $\\SL_2(\\Z)$ with coeffici
ents of size at most $q^e$? A simple pigeonhole principle shows that $e >
3/2$. In his letter Sarnak proved that this is in fact tight\, namely $e =
3/2$\, and call this optimal strong approximation for $\\SL_2(\\Z)$. The
proof relies on a density theorem of the Ramanujan conjecture for $\\SL_2(
\\Z)$.\n\nIn this talk we will give a brief overview of the strong approxi
mation\, a quantitative strengthening of it called super strong approximat
ion\, and the above mentioned optimal strong approximation phenomena\, for
arithmetic groups. We highlight the special case of $p$-arithmetic subgro
ups of classical definite matrix groups and the connection between the opt
imal strong approximation and optimal almost diameter for Ramanujan comple
xes. Finally\, we will present the Sarnak-Xue density hypothesis and descr
ibe recent ongoing works on it relying on deep results coming from the Lan
glands program.\n\nThis talk is based on ongoing joint works with B. Feigo
n\, M. Gerbelli-Gauthier\, H. Gustafssun\, K. Maurischat and O. Parzanchev
ski.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/151/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Evelina Viada (University of Göttingen)
DTSTART:20221027T150000Z
DTEND:20221027T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/152
DESCRIPTION:Title: Rational points on curves in a product of elliptic curves\nby Evelina Viada (University of Göttingen) as part of Number Theory We
b Seminar\n\n\nAbstract\nThe Mordell-Conjecture (Faltings Theorem) states
that an algebraic curve of genus at least $2$ has only finitely many ratio
nal points. The Torsion Anomalous Conjecture (TAC) generalises Faltings Th
eorem. In some cases the proofs of the TAC are effective\, implying effect
ive cases of the Mordell-Conjecture. I would like to explain an effective
method to determine the $K$-rational points on certain families of curves
and to present some new specific examples. I will give an overview of the
methods used in the context of the TAC presenting some general theorems a
nd applications.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/152/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Danny Neftin (Technion-Israel Institute of Technology)
DTSTART:20220915T150000Z
DTEND:20220915T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/153
DESCRIPTION:Title: Reducible fibers of polynomial maps\nby Danny Neftin (Te
chnion-Israel Institute of Technology) as part of Number Theory Web Semina
r\n\n\nAbstract\nFor a polynomial $f\\in \\mathbb Q[x]$\, the fiber $f^{-1
}(a)$ is irreducible over $\\mathbb Q$ for all values $a\\in \\mathbb Q$ o
utside a ``thin" set of exceptions $R_f$ whose explicit description is unk
nown in general. The problem of describing $R_f$ is closely related to red
ucibility and arboreal representations in arithmetic dynamics\, as well as
to Kronecker and arithmetic equivalence for polynomial maps\, that is\, p
olynomial versions of the question: "can you hear the shape of the drum?".
We shall discuss recent progress on the above problem and topics.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/153/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Gauthier (Université Paris-Saclay)
DTSTART:20221013T150000Z
DTEND:20221013T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/154
DESCRIPTION:Title: A complex analytic approach to sparsity\, rigidity and unifo
rmity in arithmetic dynamics\nby Thomas Gauthier (Université Paris-Sa
clay) as part of Number Theory Web Seminar\n\n\nAbstract\nThis talk is con
cerned with connections between arithmetic dynamics and complex dynamics.
The first aim of the talk is to discuss several open problems from arithme
tic dynamics and to explain how these problems are related to complex dyna
mical tool: bifurcation measures.\nIf time allows\, I will give a strategy
to tackle several of those problems at the same time. This is based on a
joint work in progress with Gabriel Vigny and Johan Taflin.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/154/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jack Thorne (University of Cambridge)
DTSTART:20221020T150000Z
DTEND:20221020T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/155
DESCRIPTION:Title: Symmetric power functoriality for $\\GL(2)$\nby Jack Tho
rne (University of Cambridge) as part of Number Theory Web Seminar\n\n\nAb
stract\nLanglands’s functoriality conjectures predict the existence of
“liftings” of automorphic representations along morphisms of $L$-group
s. A basic case of interest comes from the irreducible algebraic represent
ations of $\\GL(2)$ – the associated symmetric power $L$-functions are t
hen the ones identified by Serre in the 1960’s in relation to the Sato
—Tate conjecture.\n\nI will describe the background to these ideas and t
hen discuss the proof\, joint with James Newton\, of the existence of thes
e symmetric power liftings for Hilbert modular forms. One arithmetic conse
quence is that if $E$ is a (non-CM) elliptic curve over a real quadratic f
ield\, then all of its symmetric power $L$-functions admit analytic contin
uation to the whole complex plane.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/155/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emanuel Carneiro (ICTP)
DTSTART:20221110T160000Z
DTEND:20221110T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/156
DESCRIPTION:Title: Hilbert spaces and low-lying zeros of $L$-functions\nby
Emanuel Carneiro (ICTP) as part of Number Theory Web Seminar\n\n\nAbstract
\nIn this talk I would like to present some ideas behind a general Hilbert
space framework for solving certain optimization problems that arise when
studying the distribution of the low-lying zeros of families of $L$-funct
ions. For instance\, in connection to previous work of Iwaniec\, Luo\, and
Sarnak (2000)\, we will discuss how to use information from one-level den
sity theorems to estimate the proportion of non-vanishing of $L$-functions
in a family at a low-lying height on the critical line. We will also disc
uss the problem of estimating the height of the first low-lying zero in a
family\, considered by Hughes and Rudnick (2003) and Bernard (2015). This
is based on joint work with M. Milinovich and A. Chirre.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/156/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Trevor Wooley (Purdue University)
DTSTART:20221117T160000Z
DTEND:20221117T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/157
DESCRIPTION:Title: Waring’s Problem\nby Trevor Wooley (Purdue University)
as part of Number Theory Web Seminar\n\n\nAbstract\nIn 1770\, E. Waring m
ade an assertion these days interpreted as conjecturing that when $k$ is a
natural number\, all positive integers may be written as the sum of a num
ber $g(k)$ of positive integral $k$-th powers\, with $g(k)$ finite. Since
the work of Hardy and Littlewood a century ago\, attention has largely shi
fted to the problem of bounding $G(k)$\, the least number $s$ having the p
roperty that all sufficiently large integers can be written as the sum of
$s$ positive integral $k$-th powers. It is known that $G(2)=4$ (Lagrange)\
, $G(3)\\le 7$ (Linnik)\, $G(4)=16$ (Davenport)\, and $G(5)\\le 17$\, $G(6
)\\le 24$\, ...\, $G(20)\\le 142$ (Vaughan and Wooley). For large $k$ one
has $G(k)\\le k(\\log k+\\log \\log k+2+o(1))$ (Wooley). We report on very
recent progress joint with Joerg Bruedern. One or two new world records w
ill be on display.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/157/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Duker Lichtman (University of Oxford)
DTSTART:20221124T160000Z
DTEND:20221124T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/158
DESCRIPTION:Title: A proof of the Erdős primitive set conjecture\nby Jared
Duker Lichtman (University of Oxford) as part of Number Theory Web Semina
r\n\n\nAbstract\nA set of integers greater than 1 is primitive if no membe
r in the set divides another. Erdős proved in 1935 that the sum of 1/(a l
og a)\, ranging over a in A\, is uniformly bounded over all choices of pri
mitive sets A. In 1986 he asked if this bound is attained for the set of p
rime numbers. In this talk we describe recent work which answers Erdős’
conjecture in the affirmative. We will also discuss applications to old q
uestions of Erdős\, Sárközy\, and Szemerédi from the 1960s.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/158/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert J. Lemke Oliver (Tufts University)
DTSTART:20221201T160000Z
DTEND:20221201T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/159
DESCRIPTION:Title: Uniform exponent bounds on the number of primitive extension
s of number fields\nby Robert J. Lemke Oliver (Tufts University) as pa
rt of Number Theory Web Seminar\n\n\nAbstract\nA folklore conjecture asser
ts the existence of a constant $c_n > 0$ such that $N_n(X) \\sim c_n X$ as
$X\\to \\infty$\, where $N_n(X)$ is the number of degree $n$ extensions $
K/\\mathbb{Q}$ with discriminant bounded by $X$. This conjecture is known
if $n \\leq 5$\, but even the weaker conjecture that there exists an abso
lute constant $C\\geq 1$ such that $N_n(X) \\ll_n X^C$ remains unknown and
apparently out of reach.\n\nHere\, we make progress on this weaker conjec
ture (which we term the ``uniform exponent conjecture'') in two ways. Fir
st\, we reduce the general problem to that of studying relative extensions
of number fields whose Galois group is an almost simple group in its smal
lest degree permutation representation. Second\, for almost all such grou
ps\, we prove the strongest known upper bound on the number of such extens
ions. These bounds have the effect of resolving the uniform exponent conj
ecture for solvable groups\, sporadic groups\, exceptional groups\, and cl
assical groups of bounded rank. This is forthcoming work that grew out of
conversations with M. Bhargava.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/159/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laura DeMarco (Harvard University)
DTSTART:20221208T160000Z
DTEND:20221208T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/160
DESCRIPTION:Title: Lattès maps\, bifurcations\, and arithmetic\nby Laura D
eMarco (Harvard University) as part of Number Theory Web Seminar\n\n\nAbst
ract\nIn the field of holomorphic dynamics\, we learn that the Lattès map
s -- the rational functions on $\\mathbb{P}^1$ that are quotients of maps
on elliptic curves -- are rather boring. We can understand their dynamics
completely. But viewed arithmetically\, there are still unanswered quest
ions. I'll begin the talk with some history of these maps. Then I'll des
cribe one of the recent questions and how it has led to interesting comple
x-dynamical questions about other families of maps on $\\mathbb{P}^1$ and\
, in turn\, new perspectives on the arithmetic side. The new material is
a joint project with Myrto Mavraki.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/160/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Umberto Zannier (Scuola Normale Superiore Pisa)
DTSTART:20221222T160000Z
DTEND:20221222T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/161
DESCRIPTION:Title: Bounded generation in linear groups and exponential parametr
izations\nby Umberto Zannier (Scuola Normale Superiore Pisa) as part o
f Number Theory Web Seminar\n\n\nAbstract\nIn fairly recent joint work wit
h Corvaja\, Rapinchuk\, Ren\, we applied results from Diophantine S-unit t
heory to problems of “bounded generation” in linear groups: this prope
rty is a strong form of finite generation and is useful for several issues
in the setting. Focusing on “anisotropic groups” (i.e. containing onl
y semi-simple elements)\, we could give a simple essentially complete desc
ription of those with the property. More recently\, in further joint work
also with Demeio\, we proved the natural expectation that sets boundedly g
enerated by semi-simple elements (in linear groups over number fields) ar
e “sparse”. Actually\, this holds for all sets obtained by exponential
parametrizations. As a special consequence\, this gives back the previous
results with a different approach and additional precision and generality
.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/161/
END:VEVENT
BEGIN:VEVENT
SUMMARY:William Duke (UCLA)
DTSTART:20221215T160000Z
DTEND:20221215T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/162
DESCRIPTION:Title: On the analytic theory of isotropic ternary quadratic forms<
/a>\nby William Duke (UCLA) as part of Number Theory Web Seminar\n\n\nAbst
ract\nI will describe recent work giving an asymptotic formula for a count
of primitive integral zeros of an isotropic ternary quadratic form in an
orbit under integral automorphs of the form. The constant in the asymptoti
c is explicitly computed in terms of local data determined by the orbit.
This is compared with the well-known asymptotic for the count of all pri
mitive zeros. Together with an extension of results of Kneser by R. Schul
ze-Pillot on the classes in a genus of representations\, this yields a fo
rmula for the number of orbits\, summed over a genus of forms\, in term
s of the number of local orbits. For a certain special class of forms a si
mple explicit formula is given for this number.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/162/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Terence Tao (UCLA)
DTSTART:20230223T160000Z
DTEND:20230223T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/163
DESCRIPTION:Title: Infinite Partial Sumsets in the Primes\nby Terence Tao (
UCLA) as part of Number Theory Web Seminar\n\n\nAbstract\nIt is an open qu
estion as to whether the prime numbers contain the sum $A+B$ of two infini
te sets of natural numbers $A$\, $B$ (although results of this type are kn
own assuming the Hardy-Littlewood prime tuples conjecture). Using the May
nard sieve and the Bergelson intersectivity lemma\, we show the weaker res
ult that there exist two infinite sequences $a_1 < a_2 < ...$ and $b_1 < b
_2 < ...$ such that $a_i + b_j$ is prime for all $i < j$. Equivalently\,
the primes are not "translation-finite" in the sense of Ruppert. As an ap
plication of these methods we show that the orbit closure of the primes is
uncountable.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/163/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kannan Soundararajan (Stanford University)
DTSTART:20230406T150000Z
DTEND:20230406T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/164
DESCRIPTION:Title: Covering integers using quadratic forms\nby Kannan Sound
ararajan (Stanford University) as part of Number Theory Web Seminar\n\n\nA
bstract\nHow large must $\\Delta$ be so that we can cover a substantial pr
oportion of the integers below $X$ using the binary quadratic forms $x^2 +
dy^2$ with $d$ below $\\Delta$? Problems involving representations by bin
ary quadratic forms have a long history\, going back to Fermat. The parti
cular problem mentioned here was recently considered by Hanson and Vaughan
\, and Y. Diao. In ongoing work with Ben Green\, we resolve this problem\
, and identify a sharp phase transition: If $\\Delta$ is below $(\\log X)
^{\\log 2-\\epsilon}$ then zero percent of the integers below $X$ are repr
esented\, whereas if $\\Delta$ is above $(\\log X)^{\\log 2 +\\epsilon}$ t
hen 100 percent of the integers below $X$ are represented.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/164/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Régis de la Bretèche (Institut de Mathématiques de Jussieu-Pari
s Rive Gauche)
DTSTART:20230112T160000Z
DTEND:20230112T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/165
DESCRIPTION:Title: Higher moments of primes in arithmetic progressions\nby
Régis de la Bretèche (Institut de Mathématiques de Jussieu-Paris Rive
Gauche) as part of Number Theory Web Seminar\n\n\nAbstract\nIn a joint wor
k with Daniel Fiorilli\, we develop a new method to prove lower bounds of
some moments related to the distribution of primes in arithmetic progressi
ons. We shall present main results and explain some aspects of the proofs
. To prove our results\, we assume GRH but we succeed to avoid linearly in
dependence on zeroes hypothesis.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/165/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cecile Dartyge (Institut Élie Cartan\, Université de Lorraine)
DTSTART:20230119T160000Z
DTEND:20230119T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/166
DESCRIPTION:Title: On the largest prime factor of quartic polynomial values: th
e cyclic and dihedral cases\nby Cecile Dartyge (Institut Élie Cartan
\, Université de Lorraine) as part of Number Theory Web Seminar\n\n\nAbst
ract\nLet $P(X)$ be a monic\, quartic\, irreducible polynomial of $\\Z[X]$
with cyclic or dihedral Galois group. We prove that there exists $c_P >0$
\, such that for a positive proportion of integers $n$\, $P(n)$ has a prim
e factor bigger than $n^{1+c_P}$. This is a joint work with James Maynard.
\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/166/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Wilms (University of Basel)
DTSTART:20230126T160000Z
DTEND:20230126T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/167
DESCRIPTION:Title: On equidistribution in Arakelov theory\nby Robert Wilms
(University of Basel) as part of Number Theory Web Seminar\n\n\nAbstract\n
As a motivating example of its own interest I will first discuss a new equ
idistribution result for the zero sets of integer polynomials. More precis
ely\, I will give a condition such that the zero sets tends to equidistrib
ute with respect to the Fubini-Study measure and I will show that this con
dition is generically satisfied in sets of polynomials of bounded Bombieri
norm. In the second part\, I will embed this example in a much more gener
al framework about the distribution of the divisors of small sections of a
rithmetically ample hermitian line bundles in Arakelov theory.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/167/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Koymans (University of Michigan)
DTSTART:20230504T150000Z
DTEND:20230504T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/168
DESCRIPTION:Title: Counting nilpotent extensions\nby Peter Koymans (Univers
ity of Michigan) as part of Number Theory Web Seminar\n\n\nAbstract\nWe di
scuss some recent progress towards the strong form of Malle’s conjecture
. Even for nilpotent extensions\, only very few cases of this conjecture a
re currently known. We show how equidistribution of Frobenius elements pla
ys an essential role in this problem and how this can be used to make furt
her progress towards Malle’s conjecture. We will also discuss applicatio
ns to the Massey vanishing conjecture and to lifting problems. This is joi
nt work with Carlo Pagano.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/168/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Florian Luca (University of the Witwatersrand)
DTSTART:20230316T160000Z
DTEND:20230316T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/169
DESCRIPTION:Title: Recent progress on the Skolem problem\nby Florian Luca (
University of the Witwatersrand) as part of Number Theory Web Seminar\n\n\
nAbstract\nThe celebrated Skolem-Mahler-Lech Theorem states that the set o
f zeros of a linear recurrence sequence is the union of a finite set and f
initely many arithmetic progressions. The corresponding computational ques
tion\, the Skolem Problem\, asks to determine whether a given linear recur
rence sequence has a zero term. Although the Skolem-Mahler-Lech Theorem is
almost 90 years old\, decidability of the Skolem Problem remains open. On
e of the main contributions of the talk is to present an algorithm to solv
e the Skolem Problem for simple linear recurrence sequences (those with si
mple characteristic roots). Whenever the algorithm terminates\, it produce
s a stand-alone certificate that its output is correct -- a set of zeros t
ogether with a collection of witnesses that no further zeros exist. We giv
e a proof that the algorithm always terminates assuming two classical numb
er-theoretic conjectures: the Skolem Conjecture (also known as the Exponen
tial Local-Global Principle) and the $p$-adic Schanuel Conjecture. Prelimi
nary experiments with an implementation of this algorithm within the tool
SKOLEM point to the practical applicability of this method. \n In the seco
nd part of the talk\, we present the notion of an Universal Skolem Set\, w
hich is a subset of the positive integers on which the Skolem is decidable
regardless of the linear recurrence. We give two examples of such sets\,
one of which is of positive density (that is\, contains a positive proport
ion of all the positive integers).\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/169/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matilde Lalín (Université de Montréal)
DTSTART:20230202T160000Z
DTEND:20230202T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/170
DESCRIPTION:Title: Distributions of sums of the divisor function over function
fields\nby Matilde Lalín (Université de Montréal) as part of Number
Theory Web Seminar\n\n\nAbstract\nIn 2018 Keating\, Rodgers\, Roditty-Ger
shon and Rudnick studied the mean-square of sums of the divisor function $
d_k(f)$ over short intervals and over arithmetic progressions for the fun
ction field $\\mathbb{F}_q[T]$. By results from the Katz and Sarnak philo
sophy\, they were able to relate these problems to certain integrals over
the ensemble of unitary matrices when $q$ goes to infinity. We study simi
lar problems leading to integrals over the ensembles of symplectic and ort
hogonal matrices when $q$ goes to infinity. This is joint work with Vivian
Kuperberg.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/170/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sarah Peluse (Institute for Advanced Study and Princeton Universit
y)
DTSTART:20230209T160000Z
DTEND:20230209T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/171
DESCRIPTION:Title: Divisibility of character values of the symmetric group\
nby Sarah Peluse (Institute for Advanced Study and Princeton University) a
s part of Number Theory Web Seminar\n\n\nAbstract\nIn 2017\, Miller comput
ed the character tables of $S_n$ for all $n$ up to $38$ and looked at vari
ous statistical properties of the entries. Characters of symmetric groups
take only integer values\, and\, based on his computations\, Miller conjec
tured that almost all entries of the character table of $S_n$ are divisibl
e by any fixed prime power as $n$ tends to infinity. In this talk\, I will
discuss joint work with K. Soundararajan that resolves this conjecture\,
and mention some related open problems.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/171/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ofir Gorodetsky (University of Oxford)
DTSTART:20230323T160000Z
DTEND:20230323T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/172
DESCRIPTION:Title: How many smooth numbers and smooth polynomials are there?\nby Ofir Gorodetsky (University of Oxford) as part of Number Theory Web
Seminar\n\n\nAbstract\nSmooth numbers are integers whose prime factors are
smaller than a threshold $y$. In the 80s they became important outside of
pure math\, as Pomerance's quadratic sieve for factoring integers relied
on their distribution. The density of smooth numbers up to $x$ can be appr
oximated\, in some range\, using a peculiar function $\\rho$ called Dickma
n's function\, defined via a delay-differential equation. All of the above
is also true for smooth polynomials over finite fields.\n\nWe'll survey t
hese topics and discuss recent results concerning the range of validity of
the approximation of the density of smooth numbers by $\\rho$\, whose pro
ofs rely on relating the counting function of smooth numbers to the Rieman
n zeta function and the counting function of primes. In particular\, we un
cover phase transitions in the behavior of the density at the points $y=(\
\log x)^2$ (as conjectured by Hildebrand) and $y=(\\log x)^(3/2)$\, when p
reviously only a transition at $y=\\log x$ was known and understood. These
transitions also occur in the polynomial setting. We'll also show that a
standard conjecture on the error in the Prime Number Theorem implies $\\rh
o$ is always a lower bound for the density\, addressing a conjecture of Po
merance.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/172/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ziyang Gao (Leibniz University Hannover)
DTSTART:20230330T150000Z
DTEND:20230330T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/173
DESCRIPTION:Title: Sparsity of rational and algebraic points\nby Ziyang Gao
(Leibniz University Hannover) as part of Number Theory Web Seminar\n\n\nA
bstract\nIt is a fundamental question in mathematics to find rational solu
tions to a given system of polynomials\, and in modern language this quest
ion translates into finding rational points in algebraic varieties. This q
uestion is already very deep for algebraic curves defined over $\\Q$. An i
ntrinsic natural number associated with the curve\, called its genus\, pla
ys an important role in studying the rational points on the curve. In 1983
\, Faltings proved the famous Mordell Conjecture (proposed in 1922)\, whic
h asserts that any curve of genus at least $2$ has only finitely many rati
onal points. Thus the problem for curves of genus at least $2$ can be divi
ded into several grades: finiteness\, bound\, uniform bound\, effectivenes
s. An answer to each grade requires a better understanding of the distribu
tion of the rational points.\n\nIn my talk\, I will explain the historical
and recent developments of this problem according to the different grades
.\n\nAnother important topic on studying points on curves is the torsion p
ackets. This topic goes beyond rational points. I will also discuss briefl
y about it in my talk.\n\nIf time permits\, I will mention the correspondi
ng result in high dimensions.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/173/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wanlin Li (Université de Montréal)
DTSTART:20230216T160000Z
DTEND:20230216T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/174
DESCRIPTION:Title: Ordinary and Basic Reductions of Abelian Varieties\nby W
anlin Li (Université de Montréal) as part of Number Theory Web Seminar\n
\n\nAbstract\nGiven an abelian variety A defined over a number field\, a c
onjecture attributed to Serre states that the set of primes at which A adm
its ordinary reduction is of positive density. This conjecture had been pr
oved for elliptic curves (Serre\, 1977)\, abelian surfaces (Katz 1982\, Sa
win 2016) and certain higher dimensional abelian varieties (Pink 1983\, Fi
te 2021\, etc). \n\nIn this talk\, we will discuss ideas behind these resu
lts and recent progress for abelian varieties with non-trivial endomorphis
ms\, including some cases of A with almost complex multiplication by an ab
elian CM field\, based on joint work with Cantoral-Farfan\, Mantovan\, Pri
es\, and Tang.\n\nApart from ordinary reduction\, we will also discuss the
set of primes at which an abelian variety admits basic reduction\, genera
lizing a result of Elkies on the infinitude of supersingular primes for el
liptic curves. This is joint work with Mantovan\, Pries\, and Tang.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/174/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pierre Le Boudec (University of Basel)
DTSTART:20230427T150000Z
DTEND:20230427T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/175
DESCRIPTION:Title: $2$-torsion in class groups of number fields\nby Pierre
Le Boudec (University of Basel) as part of Number Theory Web Seminar\n\n\n
Abstract\nIt is well-known that the class number of a number field $K$ of
fixed degree $n$ is roughly bounded by the square root of the absolute val
ue of the discriminant of $K$. However\, given a prime number $p$\, the ca
rdinality of the $p$-torsion subgroup of the class group of $K$ is expecte
d to be much smaller. Unfortunately\, beating the trivial bound mentioned
above is a hard problem. Indeed\, this task had only been achieved for a h
andful of pairs $(n\,p)$ until Bhargava\, Shankar\, Taniguchi\, Thorne\, T
simerman and Zhao managed to do so for any degree $n$ in the case $p=2$. I
n this talk we will go through their proof and we will present new bounds
which depend on the geometry of the lattice underlying the ring of integer
s of $K$. This is joint work with Dante Bonolis.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/175/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Wilkie (University of Manchester)
DTSTART:20230309T160000Z
DTEND:20230309T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/176
DESCRIPTION:Title: Integer points on analytic sets\nby Alex Wilkie (Univers
ity of Manchester) as part of Number Theory Web Seminar\n\n\nAbstract\nIn
2004 I proved an $O(\\log\\log H)$ bound for the number of integer points
of height at most $H$ lying on a globally subanalytic curve. (The paper wa
s published in the Journal of Symbolic Logic and so probably escaped the n
otice of most of you reading this.) Recently\, Gareth Jones and Gal Binyam
ini proposed a generalization of the result to higher dimensions (where th
e obvious statement is almost certainly false) and I shall report on our j
oint work: one obtains the (hoped for) $(\\log\\log H)^n$ bound for (not
globally subanalytic but) globally analytic sets of dimension $n$.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/176/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandra Shlapentokh (East Carolina University)
DTSTART:20230302T160000Z
DTEND:20230302T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/177
DESCRIPTION:Title: Defining integers using unit groups\nby Alexandra Shlape
ntokh (East Carolina University) as part of Number Theory Web Seminar\n\n\
nAbstract\nWe discuss some problems of definability and decidability over
rings of integers of algebraic extensions of $\\Q$. In particular\, we sh
ow that for a large class of fields $K$ there is a simple formula defining
rational integers over $O_K$. Below $U_K$ is the group of units of $O_K$
. \n\n$\\Z=\\{x| \\forall \\varepsilon \\in U_K\\setminus \\{1\\}\\ \\exis
ts \\delta \\in U_K: x \\equiv \\frac{\\delta-1}{\\varepsilon-1} \\bmod (\
\varepsilon-1)\\}$. This talk is based on a joint paper with Barry Mazur a
nd Karl Rubin.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/177/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bjorn Poonen (MIT)
DTSTART:20230420T130000Z
DTEND:20230420T140000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/178
DESCRIPTION:Title: Integral points on curves via Baker's method and finite éta
le covers\nby Bjorn Poonen (MIT) as part of Number Theory Web Seminar\
n\n\nAbstract\nWe prove results in the direction of showing that for some
affine curves\, Baker's method applied to finite étale covers is insuffic
ient to determine the integral points.\n\nPlease note the unusual time!\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/178/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lior Bary-Soroker (Tel Aviv University)
DTSTART:20230601T150000Z
DTEND:20230601T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/179
DESCRIPTION:Title: Random additive polynomials\nby Lior Bary-Soroker (Tel A
viv University) as part of Number Theory Web Seminar\n\n\nAbstract\nRandom
polynomials with integer coefficients tend to be irreducible and to have
a large Galois group with high probability. This was shown more than a cen
tury ago in the large box model\, where we choose the coefficients uniform
ly from a box and let its size go to infinity\, while only recently there
are results in the restricted box model\, when the size of the box is boun
ded and its dimension (i.e. the degree of the polynomial) goes to infinity
. \n\nIn this talk\, we will discuss an important class of random polynomi
als — additive polynomials\, which have coefficients in the polynomial r
ing over a finite field. In this case\, the roots form a vector space\, he
nce the Galois group is naturally a subgroup of $\\GL_n$. \n\nWhile we pro
ve that the Galois group is the full matrix group both in the large box mo
del\, and in the large finite field limit\, our main result is in the rest
ricted box model: under some necessary condition the Galois group is large
(in the sense that it contains $\\SL_n$) asymptotically almost surely\, a
s the degree goes to infinity.\n\nThe proof relies crucially on deep resul
ts on subgroups of $\\GL_n$ by Fulman and Guralnick\, combined with tools
from algebra and number theory. \n\nBased on a joint work with Alexei Enti
n and Eilidh McKemmie\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/179/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ben Green (University of Oxford)
DTSTART:20230511T150000Z
DTEND:20230511T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/180
DESCRIPTION:Title: On Sarkozy's theorem for shifted primes\nby Ben Green (U
niversity of Oxford) as part of Number Theory Web Seminar\n\n\nAbstract\nS
uppose that $N$ is large and that $A$ is a subset of $\\{1\,..\,N\\}$ whic
h does not contain two elements $x\, y$ with $x - y$ equal to $p-1$\, $p$
a prime. Then $A$ has cardinality at most $N^{1 - c}$\, for some absolute
$c > 0$. I will discuss the history of this kind of question as well as so
me aspects of the proof of the stated result.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/180/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hélène Esnault (Freie Universität Berlin)
DTSTART:20230413T150000Z
DTEND:20230413T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/181
DESCRIPTION:Title: Integrality Properties of the Betti Moduli Space\nby Hé
lène Esnault (Freie Universität Berlin) as part of Number Theory Web Sem
inar\n\n\nAbstract\nWe use de Jong’s conjecture and the existence of $\\
ell$-adic companions to single out integrality properties of the Betti mod
uli space. The first such instance was in joint work with Michael Groechen
ig on Simpson’s integrality conjecture for (cohomologically) rigid local
systems. This integrality property yields an obstruction for a finitely p
resented group to be the fundamental group of a sooth quasi-projective com
plex variety. (joint with Johan de Jong)\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/181/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Shusterman (Harvard University)
DTSTART:20230518T150000Z
DTEND:20230518T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/182
DESCRIPTION:Title: Counting Minimally Ramified Global Field Extensions\nby
Mark Shusterman (Harvard University) as part of Number Theory Web Seminar\
n\n\nAbstract\nGiven a finite group $G$\, one is interested in the number
of Galois extensions of a global field with Galois group $G$ and bounded d
iscriminant. We consider a refinement of this problem where the discrimina
nt is required to have the smallest possible number of (distinct) prime fa
ctors. We will discuss existing results and conjectures over number fields
\, and present some recent results over function fields.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/182/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Barak Weiss (Tel Aviv University)
DTSTART:20230525T150000Z
DTEND:20230525T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/183
DESCRIPTION:Title: New bounds on lattice covering volumes\, and nearly uniform
covers\nby Barak Weiss (Tel Aviv University) as part of Number Theory
Web Seminar\n\n\nAbstract\nLet $L$ be a lattice in $\\R^n$ and let $K$ be
a convex body. The covering volume of $L$ with respect to $K$ is the minim
al volume of a dilate $rK$\, such that $L+rK = \\R^n$\, normalized by the
covolume of $L$. Pairs $(L\,K)$ with small covering volume correspond to e
fficient coverings of space by translates of $K$\, where the translates li
e in a lattice. Finding upper bounds on the covering volume as the dimensi
on $n$ grows is a well studied problem in the so-called “Geometry of Num
bers”\, with connections to practical questions arising in computer scie
nce and electrical engineering. In a recent paper with Or Ordentlich (EE\,
Hebrew University) and Oded Regev (CS\, NYU) we obtain substantial improv
ements to bounds of Rogers from the 1950s. In another recent paper\, we ob
tain bounds on the minimal volume of nearly uniform covers (to be defined
in the talk). The key to these results are recent breakthroughs by Dvir an
d others regarding the discrete Kakeya problem. I will give an overview of
the questions and results.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/183/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlo Pagano (Concordia University)
DTSTART:20230608T150000Z
DTEND:20230608T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/184
DESCRIPTION:Title: On Chowla's non-vanishing conjecture over function fields\nby Carlo Pagano (Concordia University) as part of Number Theory Web Sem
inar\n\n\nAbstract\nA conjecture of Chowla postulates that no $L$-function
of Dirichlet characters over the rationals vanishes at $s=1/2$. Soundarar
ajan has proved non-vanishing for a positive proportion of quadratic chara
cters. Over function fields Li has discovered that Chowla's conjecture fai
ls for infinitely many distinct quadratic characters. However\, on the bas
is of the Katz--Sarnak heuristics\, it is still widely believed that one s
hould have non-vanishing for 100% of the characters in natural families (s
uch as the family of quadratic characters). Works of Bui--Florea\, David--
Florea--Lalin\, Ellenberg--Li--Shusterman\, among others\, provided eviden
ce giving a positive proportion of non-vanishing in several such families.
I will present an upcoming joint work with Peter Koymans and Mark Shuster
man\, where we prove that for each fixed q congruent to $3$ modulo $4$ one
has 100% non-vanishing in the family of imaginary quadratic function fiel
ds.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/184/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Youness Lamzouri (Institut Elie Cartan de Lorraine)
DTSTART:20230615T150000Z
DTEND:20230615T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/185
DESCRIPTION:Title: A walk on Legendre paths\nby Youness Lamzouri (Institut
Elie Cartan de Lorraine) as part of Number Theory Web Seminar\n\n\nAbstrac
t\nIn this talk\, we shall explore certain polygonal paths\, that we call
''Legendre paths''\, which encode important information about the values o
f the Legendre symbol. More precisely\, the Legendre path modulo a prime n
umber $p$ is defined as the polygonal path in the plane whose vertices are
the points $(j\, S_p(j))$ for $0≤j≤p-1$\, where $S_p(j)$ is the (norm
alized) sum of Legendre symbols $(n/p)$ for $n$ up to $j$. In particular\
, we will attempt to answer the following questions as we vary over the pr
imes $p$: how are these paths distributed? how do their maximums behave?
when does a Legendre path decreases for the first time? what is the typica
l number of $x$-intercepts of such paths? and what proportion of a Legendr
e path is above the real axis? We will see that some of these questions c
orrespond to important and longstanding problems in analytic number theory
\, including understanding the size of the least quadratic non-residue\, a
nd improving the Pólya-Vinogradov inequality for character sums. Among ou
r results\, we prove that as we average over the primes\, the Legendre pat
hs converge in law\, in the space of continuous functions\, to a certain r
andom Fourier series constructed using Rademacher random multiplicative fu
nctions. \n\nPart of this work is joint with Ayesha Hussain and with Olek
siy Klurman and Marc Munsch.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/185/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Εfthymios Sofos (University of Glasgow)
DTSTART:20230622T150000Z
DTEND:20230622T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/186
DESCRIPTION:Title: The second moment method for rational points\nby Εfthym
ios Sofos (University of Glasgow) as part of Number Theory Web Seminar\n\n
\nAbstract\nIn a joint work with Alexei Skorobogatov we used a second-mome
nt approach to prove asymptotics for the average of the von Mangoldt funct
ion over the values of a typical integer polynomial. As a consequence\, we
proved Schinzel's Hypothesis in 100% of the cases. In addition\, we prove
d that a positive proportion of Châtelet equations have a rational point.
I will explain subsequent joint work with Tim Browning and Joni Teräväi
nen [arXiv:2212.10373] that develops the method and establishes asymptotic
s for averages of an arithmetic function over the values of typical polyno
mials. Part of the new ideas come from the theory of averages of arithmeti
c functions in short intervals. One of the applications is that the Hasse
principle holds for 100% of Châtelet equations. This agrees with the conj
ecture of Colliot-Thélène stating that the Brauer--Manin obstruction is
the only obstruction to the Hasse principle for rationally connected varie
ties.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/186/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Iosevich (University of Rochester)
DTSTART:20230629T150000Z
DTEND:20230629T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/187
DESCRIPTION:Title: Some number theoretic aspects of finite point configurations
\nby Alex Iosevich (University of Rochester) as part of Number Theory
Web Seminar\n\n\nAbstract\nWe are going to survey some recent and less rec
ent results pertaining to the study of finite point configurations in Eucl
idean space and vector spaces over finite fields\, centered around the Erd
os/Falconer distance problems. We shall place particular emphasis on numbe
r-theoretic ideas and obstructions that arise in this area.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/187/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ananth Shankar (University of Wisconsin\, Madison)
DTSTART:20231026T150000Z
DTEND:20231026T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/188
DESCRIPTION:Title: Canonical heights on Shimura varieties and the Andre-Oort co
njecture\nby Ananth Shankar (University of Wisconsin\, Madison) as par
t of Number Theory Web Seminar\n\n\nAbstract\nLet $S$ be a Shimura variety
. The Andre-Oort conjecture posits that the Zariski closure of special poi
nts must be a sub Shimura subvariety of $S$. The Andre-Oort conjecture for
$A_g$ (the moduli space of principally polarized Abelian varieties) — a
nd therefore its sub Shimura varieties — was proved by Jacob Tsimerman.
However\, this conjecture was unknown for Shimura varieties without a modu
li interpretation. Binyamini-Schmidt-Yafaev build on work of Binyamini to
reduce the Andre-Oort conjecture to establishing height bounds on special
points. I will describe joint work with Jonathan Pila and Jacob Tsimerman
where we establish these height bounds\, and therefore prove the Andre Oor
t conjecture in full generality.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/188/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Mangerel (Durham University)
DTSTART:20230907T150000Z
DTEND:20230907T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/189
DESCRIPTION:Title: Correlations\, sign patterns and rigidity theorems for multi
plicative functions\nby Alexander Mangerel (Durham University) as part
of Number Theory Web Seminar\n\n\nAbstract\nThe Liouville function $\\lam
bda(n)$\, defined to be $+1$ for $n$ having an even number of prime factor
s (counted with multiplicity) and $-1$ otherwise\, is a multiplicative fun
ction with deep connections to the distribution of primes. Inspired by the
prime $k$-tuples conjecture of Hardy and Littlewood\, Chowla conjectured
that for every $k$ each of the $2^k$ distinct sign patterns\, i.e.\, tuple
s in $\\{-1\,+1\\}^k$ are assumed by the tuples $(\\lambda(n+1)\,...\,\\la
mbda(n+k))$\, $n \\in \\mathbb{N}$\, with the same asymptotic frequency.\n
\nThe underlying phenomenon at hand is that the prime factorisations of $n
+1\,\\ldots\,n+k$ are expected to be (in a precise sense) statistically in
dependent as $n$ varies. As conjectured by Elliott\, the same equidistribu
tion of sign patterns is expected to hold for other $\\pm 1$-valued multip
licative functions\, provided they are ``far from being periodic''. To the
best of our knowledge\, until recently no explicit constructions of multi
plicative functions with this behaviour were known. \n\nIn this talk we w
ill discuss precisely what Chowla's and Elliott's conjectures say\, survey
some of the literature on correlations\, and discuss some related problem
s about sign patterns. Specifically\, we will address:\n\ni) the construct
ion of ``Liouville-like'' functions $f: \\mathbb{N} \\rightarrow \\{-1\,+1
\\}$ whose $k$-tuples $(f(n+1)\,...\,f(n+k))$ equidistribute in $\\{-1\,+1
\\}^k$\, answering a question of de la Rue from 2018\, and\n\nii) in the c
ase $k = 4$\, the classification of all $\\pm 1$-valued completely multipl
icative functions $f$ with the (rigid) property that the sequence of tuple
s $(f(n+1)\,f(n+2)\,f(n+3)\,f(n+4))$ omits the pattern $(+1\,+1\,+1\,+1)$\
, solving a 50-year old problem of R.H. Hudson.\n\nKey to these developmen
ts is a new result about the vanishing of correlations of ``moderately ape
riodic'' multiplicative functions along a dense sequence of scales.\n\nBas
ed on joint work with O. Klurman and J. Teräväinen.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/189/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Sutherland (MIT)
DTSTART:20230914T160000Z
DTEND:20230914T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/190
DESCRIPTION:Title: Murmurations of arithmetic $L$-functions\nby Andrew Suth
erland (MIT) as part of Number Theory Web Seminar\n\n\nAbstract\nWhile con
ducting a series of number-theoretic machine learning experiments last yea
r\, He\, Lee\, Oliver\, and Pozdnyakov noticed a curious oscillation in th
e averages of Frobenius traces of elliptic curves over $\\Q$. If one comp
utes the average value of $a_p(E)$ for $E/\\Q$ of fixed rank with conducto
r in a short interval\, as $p$ increases the average oscillates with a dec
aying frequency determined by the conductor. That the rank influences the
distribution of Frobenius traces has long been known (indeed\, this was t
he impetus for the experiments that led to the conjecture of Birch and Swi
nnerton-Dyer)\, but these oscillations do not appear to have been noticed
previously. This may be due in part to the critical role played by the co
nductor\; in arithmetic statistics it is common to order elliptic curves $
E/\\Q$ by naive height rather than conductor\, but doing so obscures these
oscillations.\n\nI will present results from an ongoing investigation of
this phenomenon\, which is remarkably robust and not specific to elliptic
curves. One finds similar oscillations in the averages of Dirichlet coeffi
cients of many types of $L$-functions when organized by conductor and root
number\, including those associated to modular forms and abelian varietie
s. The source of these murmurations in the case of weight-$2$ newforms wi
th trivial nebentypus is now understood\, thanks to recent work of Zubrili
na\, but all other cases remain open.\n\nThis is based on joint work with
Yang-Hui He\, Kyu-Hwan Lee\, Thomas Oliver\, and Alexey Pozdnyakov.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/190/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Timothy Browning (IST Austria)
DTSTART:20231019T150000Z
DTEND:20231019T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/191
DESCRIPTION:Title: When is a random Diophantine equation soluble over $\\mathbb
{Q}_p$ for all $p$?\nby Timothy Browning (IST Austria) as part of Numb
er Theory Web Seminar\n\n\nAbstract\nThe question in the title is of growi
ng importance in number theory and represents a more tractable staging pos
t than the question of solubility over $\\mathbb{Q}$. \nI'll describe the
landscape for various families of varieties\, which can be interpreted as
a more delicate version of Manin's conjecture\, in which one counts ration
al points of bounded height which lie in the image of adelic points under
a morphism. This leads to more subtle asymptotic behaviours and depends i
ntimately on the geometry of the morphism. This is joint work with Julian
Lyczak\, Roman Sarapin and Arne Smeets.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/191/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph H. Silverman (Brown University)
DTSTART:20231102T150000Z
DTEND:20231102T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/192
DESCRIPTION:Title: Field of Moduli and Fields of Definition in Arithmetic Geome
try and Arithmetic Dynamics\nby Joseph H. Silverman (Brown University)
as part of Number Theory Web Seminar\n\n\nAbstract\nLet $X/\\overline{\\Q
}$ be an algebraic variety defined over the field of algebraic numbers. We
say that a number field $K$ is a field of definition (FOD) for $X$ if the
re is a variety $Y/K$ such that $Y$ is $\\overline{\\Q}$-isomorphic to $X.
$\n\nThe field of moduli (FOM) of $X$ is the fixed field of\n$$\n
\\{ s \\in G_\\Q : s(X) \\textrm{ is $\\overline{\\Q}$-isomorphic to $X$}
\\}.\n$$\nIt is easy to check that every FOD for $X$ contains the FOM of $
X$\, but there are many situations where the FOM of $X$ is not a FOD. I wi
ll briefly discuss the FOM versus FOD problem in the classical case of abe
lian varieties\, and then turn to the the analogous question for morphisms
$f : \\mathbb{P}^N \\longrightarrow \\mathbb{P}^N$ defined over $\\overli
ne{\\Q}$\, where two maps are (dynamically) isomorphic if they are conjuga
te by a linear fractional transformation. I will describe what is known fo
r $N=1$\, including examples of maps for which the FOM is not an FOD. I wi
ll then discuss recent results for higher dimensional projective spaces in
which we show that every map f has a FOD whose degree over its FOM is bou
nded by a function depending only on $N$ and $\\deg(f)$. (Joint work with
John Doyle.)\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/192/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jens Marklof (University of Bristol)
DTSTART:20231109T160000Z
DTEND:20231109T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/193
DESCRIPTION:Title: Smallest denominators\nby Jens Marklof (University of Br
istol) as part of Number Theory Web Seminar\n\n\nAbstract\nIf we partition
the unit interval into $3000$ equal subintervals and take the smallest de
nominator amongst all rational points in each subinterval\, what can we sa
y about the distribution of those $3000$ denominators? I will discuss this
and related questions\, its connection with Farey statistics and random l
attices. In particular\, I will report on higher dimensional versions of a
recent proof of the 1977 Kruyswijk-Meijer conjecture by Balazard and Mart
in [Bull. Sci. Math. 187 (2023)\, Paper No. 103305] on the convergence of
the expectation value of the above distribution\, as well as closely relat
ed work by Chen and Haynes [Int. J. Number Theory 19 (2023)\, 1405--1413].
In fact\, we will uncover the full distribution and prove convergence of
more moments than just the expectation value. (This I believe was previous
ly not known even in one dimension.) We furthermore obtain a higher dimen
sional extension of Kargaev and Zhigljavsky's work on moments of the dista
nce function for the Farey sequence [J. Number Theory 65 (1997)\, 130--149
] as well as new results on pigeonhole statistics.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/193/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Henri Darmon (McGill University)
DTSTART:20231116T160000Z
DTEND:20231116T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/194
DESCRIPTION:Title: Explicit class field theory and orthogonal groups\nby He
nri Darmon (McGill University) as part of Number Theory Web Seminar\n\n\nA
bstract\nEssentially all abelian extensions of the rational numbers or of
a quadratic imaginary field\ncan be generated by special values of the exp
onential function or of the modular $j$-function\nat explicit arguments i
n the ground field. Describing the mathematical objects which could play t
he role of trigonometric and modular functions in generating class fields
of more general base fields is the stated goal of explicit class field the
ory. Around 5 years ago Jan Vonk and I proposed a framework in which clas
s fields of real quadratic fields can be generated from the special value
s of certain “rigid meromorphic cocycles” at real quadratic arguments.
Without delving into the details of this framework\, I will present some
simple concrete consequences of it in settings where the base field is tot
ally real\, and explain how they can be proved. The more general statement
s rest on (but do not require the full force of)\nthe notion of rigid mero
morphic cocycles for orthogonal groups of signature $(r\,r)$ described in
joint work with Lennart Gehrmann and Mike Lipnowski\, and are also inspir
ed by the calculations in Romain Branchereau’s PhD thesis. (Joint with
Jan Vonk)\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/194/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Isabel Vogt (Brown University)
DTSTART:20231005T150000Z
DTEND:20231005T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/195
DESCRIPTION:Title: Brauer--Manin obstructions requiring arbitrarily many Brauer
classes\nby Isabel Vogt (Brown University) as part of Number Theory W
eb Seminar\n\n\nAbstract\nA fundamental problem in the arithmetic of varie
ties over global fields is to determine whether they have a rational point
. As a first effective step\, one can check that a variety has local poin
ts for each place. However\, this is not enough\, as many classes of vari
eties are known to fail this local-global principle. The Brauer–Manin o
bstruction to the local-global principle for rational points is captured b
y elements of the Brauer group. On a projective variety\, any Brauer–Man
in obstruction is captured by a finite subgroup of the Brauer group. I wi
ll explain joint work that shows that this subgroup can require arbitraril
y many generators. This is joint with J. Berg\, C. Pagano\, B. Poonen\, M
. Stoll\, N. Triantafillou and B. Viray.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/195/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wei Zhang (MIT)
DTSTART:20231130T160000Z
DTEND:20231130T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/196
DESCRIPTION:Title: Diagonal cycles: some results and conjectures\nby Wei Zh
ang (MIT) as part of Number Theory Web Seminar\n\n\nAbstract\nAlgebraic cy
cles are among the most fundamental mathematical objects. I will discuss a
class of special algebraic cycles related to the diagonal cycle\, includi
ng the Gross-Schoen cycle (the small diagonal) on the triple product of a
curve\, the arithmetic diagonal cycle appearing in the Gan-Gross-Prasad co
njecture\, as well as the Fourier-Jacobi cycle defined by Yifeng Liu.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/196/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kaisa Matomäki (University of Turku)
DTSTART:20230921T150000Z
DTEND:20230921T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/197
DESCRIPTION:Title: Detecting primes in multiplicatively structured sequences\nby Kaisa Matomäki (University of Turku) as part of Number Theory Web S
eminar\n\n\nAbstract\nI will discuss a new sieve set-up which allows one t
o find prime numbers in sequences that have a suitable multiplicative stru
cture and good "type I information". Among other things\, the method gives
a new L-function free proof of Linnik's theorem concerning the least prim
e in an arithmetic progression. The talk is based on on-going joint work w
ith Jori Merikoski and Joni Teräväinen.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/197/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Holly Krieger (University of Cambridge)
DTSTART:20230928T150000Z
DTEND:20230928T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/198
DESCRIPTION:Title: A transcendental birational dynamical degree\nby Holly K
rieger (University of Cambridge) as part of Number Theory Web Seminar\n\n\
nAbstract\nIn the study of a discrete dynamical system defined by polynomi
als\, we wish to understand the integer sequence formed by the degrees of
the iterates of the map: examples of such a sequence include the Fibonacci
and other integer linear recurrence sequences\, but not all examples sati
sfy a finite recurrence. The growth of this sequence is measured by the d
ynamical degree\, an invariant which controls the topological\, arithmetic
\, and algebraic complexity of the system. I will discuss the surprising c
onstruction\, joint with Bell\, Diller\, and Jonsson\, of a transcendental
dynamical degree for a birational map of projective space\, and how our w
ork fits into the general phenomenon of power series taking transcendental
values at algebraic inputs.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/198/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anke Pohl (University of Bremen)
DTSTART:20231123T160000Z
DTEND:20231123T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/199
DESCRIPTION:Title: Period functions for vector-valued automorphic functions via
dynamics and cohomology\nby Anke Pohl (University of Bremen) as part
of Number Theory Web Seminar\n\n\nAbstract\nVector-valued automorphic func
tions\, or generalized automorphic functions\, occur naturally in many are
as\, most notably in spectral theory\, number theory and mathematical phys
ics. Already Selberg promoted the idea to investigate vector-valued automo
rphic functions alongside their classical relatives and to exploit their i
nteraction in order to understand their properties. While during the last
decades the focus has been on automorphic functions equivariant with regar
d to unitary representations\, the investigations recently turned to non-u
nitary representations as well. I will report on the status of an ongoing
project to investigate simultaneously unitarily and non-unitarily equivari
ant automorphic functions with a view towards period functions and a class
ical-quantum correspondence by means of dynamics (transfer operator method
s) and cohomology theory. This is joint work with R. Bruggeman.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/199/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Djordje Milićević (Bryn Mawr College)
DTSTART:20231214T160000Z
DTEND:20231214T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/200
DESCRIPTION:Title: Beyond the spherical sup-norm problem\nby Djordje Milić
ević (Bryn Mawr College) as part of Number Theory Web Seminar\n\n\nAbstra
ct\nThe sup-norm problem on arithmetic Riemannian manifolds occupies a pro
minent place at the intersection of harmonic analysis\, number theory\, an
d quantum mechanics. It asks about the sup-norm of $L^2$-normalized joint
eigenfunctions of invariant differential operators and Hecke operators —
that is\, automorphic forms — most classically in terms of their Laplac
e eigenvalues (as in the QUE problem for high-energy eigenstates)\, but al
so in terms of the volume of the manifold and other parameters.\n\nIn this
talk\, we will motivate the sup-norm problem and then describe our result
s\, joint with Blomer\, Harcos\, and Maga\, which for the first time solve
it for non-spherical Maass forms of an increasing dimension of the associ
ated $K$-type\, on an arithmetic quotient of $G=\\SL(2\,\\C)$\, with $K=\\
mathrm{SU}(2)$. We combine representation theory\, spectral analysis\, and
Diophantine arguments\, developing new Paley-Wiener theory for $G$ and sh
arp estimates on spherical trace functions of arbitrary $K$-type on the wa
y to a novel counting problem of Hecke correspondences close to various sp
ecial submanifolds of $G$.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/200/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Masser (University of Basel)
DTSTART:20231012T150000Z
DTEND:20231012T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/201
DESCRIPTION:Title: Some new elliptic integrals\nby David Masser (University
of Basel) as part of Number Theory Web Seminar\n\n\nAbstract\nIn 1981 Jam
es Davenport surmised that if an algebraic function $f(x\,t)$ is not integ
rable (with respect to $x$) by elementary means when $t$ is an independent
variable\, then there are most finitely many complex numbers $\\tau$ such
that $f(x\,\\tau)$ is integrable by elementary means. Umberto Zannier and
I in 2020 obtained a couple of counterexamples and in broad principle cla
ssified all of them with algebraic coefficients (they are necessarily some
what rare). In this talk I will review our work\, describe our recent disc
overy of entire families of the things\, and sketch an indirect connexion
with the counterexamples (known as Ribet curves) to ``relative Manin-Mumfo
rd'' found by Daniel Bertrand in 2011.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/201/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Tucker (University of Rochester)
DTSTART:20240125T160000Z
DTEND:20240125T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/202
DESCRIPTION:Title: Tits and Borel type theorems for preperiodic points of finit
e morphisms\nby Thomas Tucker (University of Rochester) as part of Num
ber Theory Web Seminar\n\n\nAbstract\nWe pose a general question: Given a
finitely generated semigroup S of finite morphisms from a variety to itsel
f\, what can one say about how the structure of the semigroup is connected
to the relationship between the preperiodic points of the elements of S?
When S consists of polarized morphisms\, we can give a fairly simple answe
r to this question using Tate's limiting procedure for Weil and Moriwaki h
eights. We formulate some conjectures that generalize this\nanswer and pr
ove some results relating to these conjectures.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/202/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zeev Rudnick (Tel Aviv University)
DTSTART:20231221T160000Z
DTEND:20231221T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/203
DESCRIPTION:Title: A talk in honor of Peter Sarnak's 70th birthday\nby Zeev
Rudnick (Tel Aviv University) as part of Number Theory Web Seminar\n\n\nA
bstract\nI will give selected highlights of Peter Sarnak's works on automo
rphic forms and some of the outstanding problems remaining.\n\nSpecial Cha
ir: Alex Kontorovich (Rutgers University)\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/203/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Misha Rudnev (University of Bristol)
DTSTART:20231207T160000Z
DTEND:20231207T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/204
DESCRIPTION:Title: The sum-product problem for integers with few prime factors<
/a>\nby Misha Rudnev (University of Bristol) as part of Number Theory Web
Seminar\n\n\nAbstract\nIt was asked by Szemerédi if the known sum-product
estimates can be improved for a set of $N$ integers under the constraint
that each integer has a small number of prime factors. We prove\, if the m
aximum number of prime factors for each integer is sub-logarithmic in $N$\
, the sum-product exponent $5/3-o(1)$. \n\nThis becomes a corollary of an
additive energy versus the product set cardinality estimate\, which turns
out to be the best possible. \nIt is based on a scheme of Burkholder-Gundy
-Davis martingale square function inequalities in $p$-adic scales\, follow
ed by an application of a variant of the Schmidt subspace theorem.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/204/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oded Regev (Courant Institute of Mathematical Sciences)
DTSTART:20240118T160000Z
DTEND:20240118T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/205
DESCRIPTION:Title: An Efficient Quantum Factoring Algorithm\nby Oded Regev
(Courant Institute of Mathematical Sciences) as part of Number Theory Web
Seminar\n\n\nAbstract\nWe show that n-bit integers can be factorized by in
dependently running a quantum circuit with $\\tilde{O}(n^{3/2})$ gates for
$\\sqrt{n}+4$ times\, and then using polynomial-time classical post-proce
ssing. In contrast\, Shor's algorithm requires circuits with $\\tilde{O}(n
^2)$ gates. The\ncorrectness of our algorithm relies on a number-theoretic
heuristic assumption reminiscent of those used in subexponential classica
l factorization algorithms. It is currently not clear if the algorithm can
lead to improved physical implementations in practice.\n\nNo background i
n quantum computation will be assumed.\n\nBased on the arXiv preprint: htt
ps://arxiv.org/abs/2308.06572\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/205/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Victor Y. Wang (IST Austria)
DTSTART:20240208T160000Z
DTEND:20240208T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/206
DESCRIPTION:Title: Sums of three cubes over a function field\nby Victor Y.
Wang (IST Austria) as part of Number Theory Web Seminar\n\n\nAbstract\nI w
ill talk about joint work with Tim Browning and Jakob Glas on producing su
ms of three cubes over a function field\, assuming a $q$-restricted form o
f the Ratios Conjecture for a geometric family of $L$-functions. If time p
ermits\, I may also discuss some recent developments in homological stabil
ity that could help to resolve this $q$-restricted Ratios Conjecture.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/206/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Akshat Mudgal (University of Oxford)
DTSTART:20240201T160000Z
DTEND:20240201T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/207
DESCRIPTION:Title: Recent progress towards the sum–product conjecture and rel
ated problems\nby Akshat Mudgal (University of Oxford) as part of Numb
er Theory Web Seminar\n\n\nAbstract\nAn important open problem in combinat
orial number theory is the Erdös–Szemerédi sum–product conjecture\,
which suggests that for any positive integers $s$\, $N$\, and for any set
$A$ of $N$ integers\, either there are many $s$-fold sums of the form $a_1
+ … + a_s$ or there are many $s$-fold products of the form $a_1…a_s$.
While this remains wide open\, various generalisations of this problem ha
ve been considered more recently\, including the question of finding large
additive and multiplicative Sidon sets in arbitrary sets of integers as w
ell as studying the so-called low energy decompositions.\n\nIn this talk\,
I will outline some recent progress towards the above questions\, as well
as highlight how these connect very naturally to other key conjectures in
additive combinatorics.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/207/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Damaris Schindler (Goettingen University)
DTSTART:20240215T160000Z
DTEND:20240215T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/208
DESCRIPTION:Title: Density of rational points near manifolds\nby Damaris Sc
hindler (Goettingen University) as part of Number Theory Web Seminar\n\n\n
Abstract\nGiven a bounded submanifold $M$ in $\\R^n$\, how many rational p
oints with common bounded denominator are there in a small thickening of $
M$? Under what conditions can we count them asymptotically as the size of
the denominator goes to infinity? I will discuss some recent work in this
direction and arithmetic applications such as Serre's dimension growth con
jecture as well as applications in Diophantine approximation. For this I'l
l focus on joint work with Shuntaro Yamagishi\, as well as joint work with
Rajula Srivastava and Niclas Technau.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/208/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paul Pollack (University of Georgia)
DTSTART:20240229T160000Z
DTEND:20240229T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/209
DESCRIPTION:Title: Stretching\, the truth about nonunique factorization\nby
Paul Pollack (University of Georgia) as part of Number Theory Web Seminar
\n\n\nAbstract\nNumber theorists learn at their mother's knee that unique
factorization fails in $\\Z[\\sqrt{-5}]$. Less well-known is that $\\Z[\\s
qrt{-5}]$ exhibits only a "half-failure" of unique factorization: while tw
o factorizations into irreducibles of the same element need not agree up t
o unit factors\, their lengths (number of factors) does always agree. This
is a special case of a 1960 result of Leonard Carlitz. I will discuss off
shoots of Carlitz's theorem. Particular attention will be paid to certain
questions of Coykendall regarding "elasticity" of orders in quadratic numb
er fields.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/209/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jacob Tsimerman (University of Toronto)
DTSTART:20240509T150000Z
DTEND:20240509T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/210
DESCRIPTION:Title: Large Compact Subvarieties of $A_g$\nby Jacob Tsimerman
(University of Toronto) as part of Number Theory Web Seminar\n\n\nAbstract
\n(Joint with Samuel Grushevsky\, Gabriele Mondello\, Riccardo Salvati Man
ni) We determine the maximal dimension of a compact subvariety of the modu
li space of principally polarized abelian varieties $A_g$ for any value of
$g$. For $g<16$ the dimension is $g-1$\, while for $g\\ge 16$\, it is det
ermined by the larged dimensional compact shimura subvariety\, which we de
termine. Our methods use functional transcendence theory.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/210/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arul Shankar (University of Toronto)
DTSTART:20240222T160000Z
DTEND:20240222T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/211
DESCRIPTION:Title: Secondary terms in the first moment of the 2-Selmer groups o
f elliptic curves\nby Arul Shankar (University of Toronto) as part of
Number Theory Web Seminar\n\n\nAbstract\nA web of interrelated conjectures
(due to work of Goldfeld\, Katz--Sarnak\, Poonen-Rains\, Bhargava--Kane--
Lenstra--Poonen--Rains) predict the distributions of ranks and Selmer grou
ps of elliptic curves over $\\Q$. These conjectures predict that the avera
ge rank of elliptic curves is $1/2$. Furthermore\, it is known (due to Bha
rgava and myself) that the average size of the $2$-Selmer group of ellipti
c curves is $3$ (when the family of all elliptic curves is ordered by (nai
ve) height). \n\nOn the computational side\, Balakrishnan\, Ho\, Kaplan\,
Spicer\, Stein\, and Weigand collect and analyze data on ranks\, $2$-Selme
r groups\, and other arithmetic invariants of elliptic curves\, when order
ed by height. Interestingly\, they find both a larger average rank as well
as a smaller average size of the $2$-Selmer group in the data. In this t
alk\, we will discuss joint work with Takashi Taniguchi\, in which we give
a possible theoretical explanation for deviation of the data on $2$-Selme
r groups from the predicted distribution\, namely\, the existence of a sec
ondary term.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/211/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marc Munsch (Jean Monnet University)
DTSTART:20240314T160000Z
DTEND:20240314T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/212
DESCRIPTION:Title: Two tales on quadratic character sums\nby Marc Munsch (J
ean Monnet University) as part of Number Theory Web Seminar\n\n\nAbstract\
nIn this talk\, we report progress on two questions on sums of real Dirich
let characters. \n\nFirstly\, we discuss quantitative results about the n
umber of sign changes in the partial sums of the real character \n$\\chi_D
$. Our method allows us to locate these changes on a very short initial in
terval (which goes beyond the range in Vinogradov's conjecture for the lea
st quadratic non-residue). The flexibility or our method allows us to dedu
ce similar results in the case \nof random multiplicative functions. \n\n
These results are related with the location of real zeros of Fekete polyno
mials $F_D$\, namely the polynomials whose coefficients are the values of
\nthe real character $\\chi_D$. \n\nIn a second part\, we will consider e
xponential sums $\\sum_{n\\le D} \\chi_D(n) e(n\\theta)$ (in other words F
ekete polynomial on the unit circle).\nRecently Harper showed that the res
tricted sum up to $H$ converges (after normalization) to the standard comp
lex Gaussian \nwhen both $\\chi_D$ and $\\theta\\in [0\,1]$ are selected u
niformly at random and $H$ is small enough. We prove that\nthe distributi
on of the values of Fekete polynomials on the unit circle is very differen
t and is governed by an explicit limiting (non-Gaussian) random point proc
ess. As an application\, we solve an open problem about the Mahler measure
of $F_p$ as $p \\rightarrow +\\infty$. \n\nThis is based on joint works
with Oleksiy Klurman and Youness Lamzouri.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/212/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vesselin Dimitrov (Caltech)
DTSTART:20240307T160000Z
DTEND:20240307T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/213
DESCRIPTION:Title: The next case after Apéry on mixed Tate periods\nby Ves
selin Dimitrov (Caltech) as part of Number Theory Web Seminar\n\n\nAbstrac
t\nI will introduce a method\, joint with Frank Calegari and Yunqing Tang\
, for proving linear independence results and effective bad approximabilit
y measures. It is an outgrowth of our previous joint work on the so-called
"unbounded denominators conjecture\," which was in some sense an applicat
ion of transcendental number theory to modular forms theory\, with the key
step being to prove sufficiently sharp $\\mathbb{Q}(x)$-linear dimension
bounds on certain spaces of algebraic functions. This time\, we step into
the wilder realm of G-functions with infinite monodromy\, and devise holon
omy bounds fine enough to prove the linear independence of two certain Dir
ichlet L-function values\, a result that\, in the realm of mixed Tate peri
ods\, can be considered as the next-simplest case after Apery's proof of t
he irrationality of $\\zeta(3)$ (excluding the cases that reduce to the He
rmite--Lindemann theorem or the Gelfond--Baker theorem on linear forms in
logarithms). One key input turns out to be the classical Shidlovsky lemma
on functional bad approximability\, the point Siegel missed for three deca
des to complete his theory of algebraic relations among special values of
E-functions. \n\nThis is all a joint work with Frank Calegari and Yunqing
Tang.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/213/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Javier Fresán (Sorbonne University)
DTSTART:20240321T160000Z
DTEND:20240321T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/214
DESCRIPTION:Title: E-functions and Geometry\nby Javier Fresán (Sorbonne Un
iversity) as part of Number Theory Web Seminar\n\n\nAbstract\nE-functions
are power series which solve a differential equation and whose coefficient
s are algebraic numbers that satisfy certain growth conditions of arithmet
ic nature. They were introduced in Siegel's 1929 memoir on the application
s of diophantine approximation with the goal of generalising the Hermite--
Lindemann--Weierstrass theorem about the transcendence of the values of th
e exponential function at algebraic arguments. Besides the exponential\, s
tandard examples include the Bessel function and confluent hypergeometric
series. After briefly surveying on the history of E-functions\, I will pre
sent a joint work in progress with Peter Jossen where we prove that expone
ntial period functions provide us with a rich geometric source of E-functi
ons. The easiest examples\, attached to polynomials of degree 4\, already
allowed us a couple of years ago to exhibit some E-functions which are not
polynomial expressions in hypergeometric series\, thus solving one of the
problems in Siegel's original paper.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/214/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nina Zubrilina (Princeton University)
DTSTART:20240516T150000Z
DTEND:20240516T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/215
DESCRIPTION:Title: Murmurations of modular forms\nby Nina Zubrilina (Prince
ton University) as part of Number Theory Web Seminar\n\n\nAbstract\nIn a r
ecent machine learning-based study\, He\, Lee\, Oliver\, and Pozdnyakov ob
served an unexpected oscillating pattern in the average value of the $P$-t
h Frobenius trace of elliptic curves of prescribed rank and conductor in a
n interval range. Sutherland later discovered that this bias extends to Di
richlet coefficients of other classes of $L$-functions when split by root
number. In my talk\, I will prove this bias for a family of holomorphic mo
dular forms and for a family of Maass forms.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/215/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dan Petersen (Stockholm University)
DTSTART:20240404T150000Z
DTEND:20240404T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/216
DESCRIPTION:Title: Moments of families of quadratic $L$-functions over function
fields via homotopy theory\nby Dan Petersen (Stockholm University) as
part of Number Theory Web Seminar\n\n\nAbstract\nThis is a report of join
t work with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams
. There is a "recipe" due to Conrey-Farmer-Keating-Rubinstein-Snaith which
allows for precise predictions for the asymptotics of moments of many dif
ferent families of $L$-functions. Our work concerns the CFKRS predictions
in the case of the quadratic family over function fields\, i.e. the family
of all $L$-functions attached to hyperelliptic curves over some fixed fin
ite field. One can relate this problem to understanding the homology of th
e braid group with certain symplectic coefficients. With Bergström-Diacon
u-Westerland we compute the stable homology groups of the braid groups wit
h these coefficients\, together with their structure as Galois representat
ions. We moreover show that the answer matches the number-theoretic predic
tions. With Miller-Patzt-Randal-Williams we prove a uniform range for homo
logical stability with these coefficients. Together\, these results imply
the CFKRS predictions for all moments in the function field case\, for all
sufficiently large (but fixed) $q$.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/216/
END:VEVENT
BEGIN:VEVENT
SUMMARY:John Voight (Dartmouth College)
DTSTART:20240523T150000Z
DTEND:20240523T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/217
DESCRIPTION:Title: The Bezout identity and norms from a quadratic extension
\nby John Voight (Dartmouth College) as part of Number Theory Web Seminar\
n\n\nAbstract\nGiven coprime integers $a\,b$\, a classical identity provid
es integers $u\,v$ such that $au-bv = 1$. We consider refinements to this
identity\, where we ask that $u\,v$ are norms from a quadratic extension.
This is joint work with Donald Cartwright and Xavier Roulleau.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/217/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yiannis Petridis (University College London)
DTSTART:20240411T150000Z
DTEND:20240411T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/218
DESCRIPTION:Title: Counting and equidistribution\nby Yiannis Petridis (Univ
ersity College London) as part of Number Theory Web Seminar\n\n\nAbstract\
nI will discuss how counting orbits in hyperbolic spaces lead to interesti
ng number theoretic problems. The counting problems (and the associated eq
uidistribution) can be studied with various methods\, and I will emphasize
automorphic form techniques\, originating in the work of H. Huber and stu
died extensively by A. Good. My collaborators in various aspects of this p
roject are Chatzakos\, Lekkas\, Risager\, and Voskou.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/218/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Theresa Anderson (Carnegie Mellon University)
DTSTART:20240425T150000Z
DTEND:20240425T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/219
DESCRIPTION:Title: Counting with new tools\nby Theresa Anderson (Carnegie M
ellon University) as part of Number Theory Web Seminar\n\n\nAbstract\nArit
hmetic statistics\, or the counting of objects of algebraic interest\, has
seen a lot of development in the last twenty years. We will take a glimps
e into just a few recent advances\, with an emphasis on the wide interplay
of new tools and techniques.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/219/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rainer Dietmann (Royal Holloway\, University of London)
DTSTART:20240530T150000Z
DTEND:20240530T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/220
DESCRIPTION:Title: Longer gaps between values of binary quadratic forms\nby
Rainer Dietmann (Royal Holloway\, University of London) as part of Number
Theory Web Seminar\n\n\nAbstract\nIt is not hard to show that there are i
nfinitely many pairs of consecutive integers that are sums of two squares.
The question about large gaps between sums of two squares is much more di
fficult. In this talk I want to report on recent joint work with Christian
Elsholtz\, Alexander Kalmynin\, Sergei Konyagin and James Maynard which m
akes progress on this and related problems\, in particular improving an ol
d record of Richards from 1982.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/220/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Philippe Michel (EPFL)
DTSTART:20240613T150000Z
DTEND:20240613T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/221
DESCRIPTION:Title: Mixed moments for Dirichlet L-functions\nby Philippe Mic
hel (EPFL) as part of Number Theory Web Seminar\n\n\nAbstract\nIn this tal
k we discuss the problem of evaluating somewhat exotic moments of Dirichle
t L-functions of large modulus (called « mixed »).\n\nNamely moments of
the shape\n\n$$\\sum_{\\chi(q)} L(\\chi^{a_1}\,1/2)\\cdots L(\\chi^a_k\,1/
2)$$\n\nwhere $q$ is a growing prime and $a_i\,\\ 1\\leq i\\leq k$ are fix
ed integers (that are not necessarily equal nor equal to $\\pm 1$).\n\nWe
will discuss some partial results focusing mainly on the case $k=2$ and $3
$.\nThe techniques involved are non trivial bounds for solutions to monom
ial congruences equations as well as for averages of hyper-Kloosterman sum
s in short intervals.\n\nThis is joint work with E. Fouvry\, E. Kowalski a
nd W. Sawin.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/221/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wouter Castryck (KU Leuven)
DTSTART:20240328T170000Z
DTEND:20240328T180000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/222
DESCRIPTION:Title: The isogeny interpolation problem\nby Wouter Castryck (K
U Leuven) as part of Number Theory Web Seminar\n\n\nAbstract\nIt is easy t
o prove that a degree-$d$ isogeny $f$ between two elliptic curves $E$ and
$E'$ is completely determined by the images of any $4d + 1$ points. In thi
s talk we will study the algorithmic problem of evaluating $f$ at a given
point $P$ on $E$\, merely upon input of such "interpolation data". In case
the interpolation points generate a group containing $E[N]$ such that $N^
2 > 4d$ is smooth and coprime to $d$ and the field characteristic\, this p
roblem was solved in 2022 by Robert\, in the context of breaking SIKE (= S
IDH)\, a former candidate for post-quantum key exchange that had advanced
to the final stage of a standardization effort run by the National Institu
te of Standards and Technology. We will discuss this solution\, and then s
how how to address more general instances of the isogeny interpolation pro
blem\, while also publicizing some unsolved cases.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/222/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Uri Shapira (Technion – Israel Institute of Technology)
DTSTART:20240418T150000Z
DTEND:20240418T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/223
DESCRIPTION:Title: Distribution of conditional directional lattices\nby Uri
Shapira (Technion – Israel Institute of Technology) as part of Number T
heory Web Seminar\n\n\nAbstract\nGiven an integral vector $v$ in Euclidean
$n$-space we project the standard lattice $\\Z^n$ into the hyperplane ort
hogonal to $v$ and obtain in this manner a "lattice of rank $n-1$" in that
hyperplane\, which is called "The directional lattice $D(\\Z^n\,v)$". \n\
nIn this talk I will discuss results about the limit distribution of direc
tional lattices as we let the vector $v$ vary in some natural sets from a
number theoretic point of view. These include\, balls\, spheres\, non-comp
act quadratic surfaces\, and integral vectors approximating an irrational
line.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/223/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chris Daw (University of Reading)
DTSTART:20240627T150000Z
DTEND:20240627T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/224
DESCRIPTION:Title: Large Galois orbits under multiplicative degeneration\nb
y Chris Daw (University of Reading) as part of Number Theory Web Seminar\n
\n\nAbstract\nThe Pila-Zannier strategy is a powerful technique for provin
g results in unlikely intersections. In this talk\, I will recall the Zilb
er-Pink conjecture for Shimura varieties and describe how Pila-Zannier wor
ks in this setting. I will highlight the most difficult outstanding obstac
le to implementing the strategy — the so-called Large Galois Orbits conj
ecture — and I will explain recent progress towards this conjecture\, bu
ilding on the works of André and Bombieri. This is joint with Martin Orr
(Manchester).\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/224/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jordan Ellenberg (University of Wisconsin–Madison)
DTSTART:20240926T150000Z
DTEND:20240926T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/225
DESCRIPTION:Title: What does machine learning have to offer number theory?\
nby Jordan Ellenberg (University of Wisconsin–Madison) as part of Number
Theory Web Seminar\n\n\nAbstract\nThis is going to be a somewhat informal
where I report on some of my own work\, some work of others\, and some st
uff I’m learning about at the ongoing Harvard/CMSA workshop on machine l
earning in mathematics. I will focus on an outlook where the goal is not
to reproduce or replace our central enterprise of writing proofs of theore
ms and understanding things\, but rather on models for machine-human colla
boration\, where ML techniques are used to generate interesting hypotheses
\, examples\, and ideas as a kind of force multiplier for traditional math
ematicians. I’ll probably talk about cap sets\, computing GCDs\, murmur
ations\, navigating Cayley graphs\, and probably some other stuff besides!
(Note: Oct 7-11 will be a number theory week at CMSA so any questions t
he audience wants to suggest we work on there are very welcome!)\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/225/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dang-Khoa Nguyen (University of Calgary)
DTSTART:20240606T150000Z
DTEND:20240606T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/226
DESCRIPTION:Title: The Pólya-Carlson dichotomy for some dynamical zeta functio
ns\nby Dang-Khoa Nguyen (University of Calgary) as part of Number Theo
ry Web Seminar\n\n\nAbstract\nLet $\\theta$ be a map from a set $X$ to its
elf. Suppose that for $k\\geq 1$\, the number $N_k(\\theta)$ of fixed poin
ts of the $k$-th fold iterate $\\theta^k=\\theta\\circ\\cdots\\circ\\theta
$ is finite. Then we can define the dynamical or Artin-Mazur zeta function
\n$$\\zeta_{\\theta}(z)=\\exp\\left(\\sum_{k=1}^{\\infty}\\frac{N_k(\\thet
a)}{k}z^k\\right).$$\nA complex power series with radius of convergence $R
\\in (0\,\\infty)$ is said to satisfy the P\\'olya-Carlson dichotomy if it
is either a rational function or it cannot be extended analytically beyon
d the disk of radius $R$.\n\nIn this talk\, we discuss the Pólya-Carlson
dichotomy for the Artin-Mazur zeta functions of endomorphisms of tori and
abelian varieties. This is from a joint work with Bell\, Gunn\, and Saunde
rs and another with Baril Boudreau and Holmes.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/226/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashwin Sah (MIT)
DTSTART:20240502T150000Z
DTEND:20240502T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/227
DESCRIPTION:Title: Quasipolynomial bounds on the inverse theorem for the Gowers
norms and applications\nby Ashwin Sah (MIT) as part of Number Theory
Web Seminar\n\n\nAbstract\nRecent work\, joint with James Leng and Mehtaab
Sawhney\, improves the so-called “inverse theorem” for the Gowers $U^
{s+1}[N]$-norm which arises in the field of additive combinatorics in rela
tion to Roth’s and Szemerédi’s theorems. I will explain how the field
of higher-order Fourier analysis broadly extends Fourier methods and the
circle method in number theory\, and discuss implications of bounds for in
verse theorems.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/227/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Max Wenqiang Xu (Stanford University)
DTSTART:20240620T150000Z
DTEND:20240620T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/228
DESCRIPTION:Title: Real zeros of Fekete polynomials and positive definite chara
cters\nby Max Wenqiang Xu (Stanford University) as part of Number Theo
ry Web Seminar\n\n\nAbstract\nIn 1911\, Fekete proposed the problem of stu
dying how likely a Fekete polynomial has no real zeros in $[0\,1]$. The wo
rk of Baker and Montgomery in 1989 qualitatively showed that Fekete polyno
mials without real zeros in $[0\,1]$ are rare. A closely related question
is asking how likely a quadratic character has nonnegative partial sums at
any stopping point. In a joint work (in progress) with Angelo and Soundar
arajan\, we give a quantitative upper bound which is close to the conjectu
ral bound.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/228/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Maynard (University of Oxford)
DTSTART:20241010T150000Z
DTEND:20241010T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/229
DESCRIPTION:Title: On the theory of prime producing sieves\, part 1\nby Jam
es Maynard (University of Oxford) as part of Number Theory Web Seminar\n\n
\nAbstract\nThe closest thing to a general method for counting primes in a
set is the method of Type I/II sums. This allows one to obtain an asympto
tic formula (or perhaps a non-trivial lower bound) for the number of prime
s in the set\, provided one has sufficiently good estimates for certain au
xiliary sums.\n\nUnfortunately what counts as 'sufficiently good' is poorl
y understood\, as are the limits of this approach. In this talk\, I'll dis
cuss a new framework (joint with Kevin Ford) which allows us to prove nece
ssary and sufficient conditions in various cases\, focusing on general fea
tures and illustrating the method with some simple examples.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/229/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rachel Newton (King’s College London)
DTSTART:20240905T150000Z
DTEND:20240905T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/230
DESCRIPTION:Title: Evaluating the wild Brauer group\nby Rachel Newton (King
’s College London) as part of Number Theory Web Seminar\n\n\nAbstract\nT
he local-global approach to the study of rational points on varieties over
number fields begins by embedding the set of rational points on a variety
$X$ into the set of its adelic points. The Brauer--Manin pairing cuts out
a subset of the adelic points\, called the Brauer--Manin set\, that conta
ins the rational points. If the set of adelic points is non-empty but the
Brauer--Manin set is empty then we say there's a Brauer--Manin obstruction
to the existence of rational points on $X$. Computing the Brauer--Manin p
airing involves evaluating elements of the Brauer group of $X$ at local po
ints. If an element of the Brauer group has order coprime to $p$\, then it
s evaluation at a $p$-adic point factors via reduction of the point modulo
$p$. For $p$-torsion elements this is no longer the case: in order to com
pute the evaluation map one must know the point to a higher $p$-adic preci
sion. Classifying Brauer group elements according to the precision require
d to evaluate them at $p$-adic points gives a filtration which we describe
using work of Bloch and Kato. Applications of our work include addressing
Swinnerton-Dyer's question about which places can play a role in the Brau
er--Manin obstruction. This is joint work with Martin Bright.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/230/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dubi Kelmer (Boston College)
DTSTART:20241107T160000Z
DTEND:20241107T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/231
DESCRIPTION:Title: Values of quadratic forms at integer points\nby Dubi Kel
mer (Boston College) as part of Number Theory Web Seminar\n\n\nAbstract\nT
he Oppenheim conjecture\, proved by Margulis\, states that the values at i
ntegers of an indefinite irrational quadratic form in $3$ or more variable
s are dense on the real line.\nIn this talk I will survey some recent resu
lts regarding effectiveness of this result for homogenous as well as inhom
ogeneous forms.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/231/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ilya Shkredov (Purdue University)
DTSTART:20241003T150000Z
DTEND:20241003T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/232
DESCRIPTION:Title: Higher Sumsets and Energies in Additive Combinatorics and Nu
mber Theory\nby Ilya Shkredov (Purdue University) as part of Number Th
eory Web Seminar\n\n\nAbstract\nWe provide an overview of the results obta
ined by the method of higher sumsets and higher energies to some problems
of additive combinatorics (the sum—product phenomenon and incidence geom
etry\, universality\, additive decomposition\, etc.)\, number theory (expo
nential sums over subgroups and Gauss sums\, sums with multiplicative char
acters\, the square—root barrier)\, Fourier analysis (uncertainty princi
ple) and others. We will also discuss some perspectives for this approach.
\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/232/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niclas Technau (University of Bonn)
DTSTART:20241024T150000Z
DTEND:20241024T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/233
DESCRIPTION:Title: Smooth discrepancy and Littlewood’s conjecture\nby Nic
las Technau (University of Bonn) as part of Number Theory Web Seminar\n\n\
nAbstract\nLet $\\boldsymbol \\alpha \\in [0\,1]^d$. This talk concerns fi
ne-scale statistics of the Kronecker sequences $(n \\boldsymbol \\alpha \\
: \\mathrm{mod} \\: 1)_{n=1}^\\infty$.\nReporting on joint work with Sam C
how\, I will discuss a local-to-global principle. The principle relates th
e smooth discrepancy\n(a global\, analytic quantity) of Kronecker sequence
s to their multiplicative diophantine approximability (a local\, arithmeti
c property).\nThis opens up a new avenue of attack for a conjecture of Lit
tlewood.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/233/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anders Södergren (Chalmers University of Technology)
DTSTART:20241114T160000Z
DTEND:20241114T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/234
DESCRIPTION:Title: Low-lying zeros in families of modular form $L$-functions\nby Anders Södergren (Chalmers University of Technology) as part of Num
ber Theory Web Seminar\n\n\nAbstract\nIn this talk\, I will discuss the di
stribution of zeros in families of $L$-functions. The focus will be on ide
as and results related to the Katz-Sarnak heuristic for the statistics of
low-lying zeros\, that is\, zeros that are located close to the real axis.
In particular\, I will report on joint work with Martin Čech\, Lucile De
vin\, Daniel Fiorilli and Kaisa Matomäki on extended density theorems in
certain families of $L$-functions attached to holomorphic modular forms or
Maass forms.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/234/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katherine E. Stange (University of Colorado\, Boulder)
DTSTART:20240919T150000Z
DTEND:20240919T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/235
DESCRIPTION:Title: The local-global conjecture for Apollonian circle packings i
s false - CANCELLED\nby Katherine E. Stange (University of Colorado\,
Boulder) as part of Number Theory Web Seminar\n\n\nAbstract\nPrimitive int
egral Apollonian circle packings are fractal arrangements of tangent circl
es with integer curvatures. The curvatures form an orbit of a 'thin group
\,' a subgroup of an algebraic group having infinite index in its Zariski
closure. The curvatures that appear must fall into one of six or eight re
sidue classes modulo $24$. The twenty-year-old local-global conjecture sta
tes that every sufficiently large integer in one of these residue classes
will appear as a curvature in the packing. We prove that this conjecture i
s false for many packings\, by proving that certain quadratic and quartic
families are missed. The new obstructions are a property of the thin Apoll
onian group (and not its Zariski closure)\, and are a result of quadratic
and quartic reciprocity\, reminiscent of a Brauer-Manin obstruction. Based
on computational evidence\, we formulate a new conjecture. This is joint
work with Summer Haag\, Clyde Kertzer\, and James Rickards. Time permitt
ing\, I will discuss some new results\, joint with Rickards\, that extend
these phenomena to certain settings in the study of continued fractions.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/235/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mehtaab Sawhney (Columbia University)
DTSTART:20240912T150000Z
DTEND:20240912T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/236
DESCRIPTION:Title: Primes of the form $p^2 + nq^2$\nby Mehtaab Sawhney (Col
umbia University) as part of Number Theory Web Seminar\n\n\nAbstract\nSupp
ose that $n$ is $0$ or $4 \\mod 6$. We show that there are infinitely many
primes of the form $p^2 + nq^2$ with both $p$ and $q$ prime\, and obtain
an asymptotic for their number. In particular\, when $n = 4$ we verify the
`Gaussian primes conjecture' of Friedlander and Iwaniec.\nJoint w. Ben Gr
een (Oxford)\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/236/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manfred Einsiedler (ETH Zürich)
DTSTART:20241031T160000Z
DTEND:20241031T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/237
DESCRIPTION:Title: Effective Equidistribution of semisimple adelic periods\
nby Manfred Einsiedler (ETH Zürich) as part of Number Theory Web Seminar\
n\n\nAbstract\nWe will discuss an effective equidistribution theorem for s
emisimple closed orbits on compact adelic quotients\, obtained in ongoing
joint work with E. Lindenstrauss\, A. Mohammadi\, and A. Wieser. The obtai
ned error depends polynomially on the minimal complexity of intermediate o
rbits and the complexity of the ambient space. The proof uses dynamical ar
guments\, Clozel's property (tau)\, Prasad's volume formula\, an effective
closing lemma\, and a novel effective generation result for subgroups. Th
e latter in turn relies on an effective version of Greenberg's theorem.\n\
nWe apply the above to the problem of establishing a local-global principl
e for representations of quadratic forms\, improving the codimension assum
ptions and providing effective bounds in a theorem of Ellenberg and Venkat
esh.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/237/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martin Orr (University of Manchester)
DTSTART:20241121T160000Z
DTEND:20241121T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/238
DESCRIPTION:Title: Height bounds for very unlikely intersections in abelian var
ieties using $G$-functions\nby Martin Orr (University of Manchester) a
s part of Number Theory Web Seminar\n\n\nAbstract\nA special case of the Z
ilber-Pink conjecture\, proved by Habegger and Pila\, states that a generi
c curve $C$ in an abelian variety $A$ has only finitely many "unlikely int
ersections"\, that is\, intersections of $C$ with subgroups of $A$ of codi
mension at least $2$. One important ingredient in the proof is a bound fo
r the height of these intersection points. In this talk\, I will discuss
a new method of proving such a height bound for intersections with subgrou
ps of large codimension ("very unlikely intersections")\, using ideas of B
ombieri and André about $G$-functions. A benefit of this method is that
it is in principle effective.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/238/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Asif Zaman (University of Toronto)
DTSTART:20241205T160000Z
DTEND:20241205T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/239
DESCRIPTION:Title: The least prime in the Chebotarev density theorem for symmet
ric groups and more\nby Asif Zaman (University of Toronto) as part of
Number Theory Web Seminar\n\n\nAbstract\nLet $K/k$ be a Galois extension o
f number fields with Galois group $G$. For a conjugacy class $C$ of $G$\,
the least unramified prime with Frobenius element in $C$ is known to be at
most a fixed absolute power of the discriminant of $K$ due to the celebra
ted work of Lagarias\, Montgomery\, and Odlyzko (1979). This theorem has b
een extensively studied with the primary method exploiting statistics of z
eros of L-functions. The current record for the exponent is 16 due to Kadi
ri\, Ng\, and Wong (2019). For $G = S_n$\, I will describe a method based
on detecting sign changes that improves this exponent to decay exponential
ly with $n$ as $n \\to \\infty$. The ideas also apply to other groups $G$
and conjugacy invariant subsets $C$.\n\nThis talk is based on joint work
with Peter Cho and Robert Lemke Oliver.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/239/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin Ford (University of Illinois at Urbana-Champaign)
DTSTART:20241017T150000Z
DTEND:20241017T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/240
DESCRIPTION:Title: On the theory of prime producing sieves\, part 2\nby Kev
in Ford (University of Illinois at Urbana-Champaign) as part of Number The
ory Web Seminar\n\n\nAbstract\nThe closest thing to a general method for c
ounting primes in a set is the method of Type I/II sums. This allows one t
o obtain an asymptotic formula (or perhaps a non-trivial lower bound) for
the number of primes in the set\, provided one has sufficiently good estim
ates for certain auxiliary sums.\n\nUnfortunately what counts as 'sufficie
ntly good' is poorly understood\, as are the limits of this approach. In t
his talk I'll talk a new framework (joint with James Maynard) which allows
us to prove various necessary and sufficient conditions\, focusing on met
hods for constructing sets that satisfy the Type I and Type II bounds yet
contain no primes. In particular\, I will go into some detail about how t
o prove that a substantial 'Type II range' is necessary to deduce the exis
tence of primes in a set.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/240/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julian Demeio (University of Bath)
DTSTART:20241128T160000Z
DTEND:20241128T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/241
DESCRIPTION:Title: The Grunwald Problem for solvable groups\nby Julian Deme
io (University of Bath) as part of Number Theory Web Seminar\n\n\nAbstract
\nLet $K$ be a number field. The Grunwald problem for a finite group (sche
me) G/K asks what is the closure of the image of $H^1(K\,G) \\to \\prod_{v
\\in M_K} H^1(K_v\,G)$. For a general $G$\, there is a Brauer—Manin obs
truction to the problem\, and this is conjectured to be the only one. In 2
017\, Harpaz and Wittenberg introduced a technique that managed to give a
positive answer (BMO is the only one) for supersolvable groups. I will pre
sent a new fibration theorem over quasi-trivial tori that\, combined with
the approach of Harpaz and Wittenberg\, gives a positive answer for all so
lvable groups.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/241/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katherine E. Stange (University of Colorado\, Boulder)
DTSTART:20250123T160000Z
DTEND:20250123T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/242
DESCRIPTION:Title: The local-global conjecture for Apollonian circle packings i
s false\nby Katherine E. Stange (University of Colorado\, Boulder) as
part of Number Theory Web Seminar\n\n\nAbstract\nPrimitive integral Apollo
nian circle packings are fractal arrangements of tangent circles with inte
ger curvatures. The curvatures form an orbit of a 'thin group'\, a subgro
up of an algebraic group having infinite index in its Zariski closure. Th
e curvatures that appear must fall into a restricted class of residues mod
ulo 24. The twenty-year-old local-global conjecture states that every suff
iciently large integer in one of these residue classes will appear as a cu
rvature in the packing. We prove that this conjecture is false for many pa
ckings\, by proving that certain quadratic and quartic families are missed
. The new obstructions are a property of the thin Apollonian group (and no
t its Zariski closure)\, and are a result of quadratic and quartic recipro
city\, reminiscent of a Brauer-Manin obstruction. Based on computational e
vidence\, we formulate a new conjecture. This is joint work with Summer H
aag\, Clyde Kertzer\, and James Rickards. Time permitting\, I will discus
s some new results\, joint with Rickards\, that extend these phenomena to
certain settings in the study of continued fractions.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/242/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katharine Woo (Princeton University)
DTSTART:20241212T160000Z
DTEND:20241212T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/243
DESCRIPTION:Title: Manin's conjecture for Châtelet surfaces\nby Katharine
Woo (Princeton University) as part of Number Theory Web Seminar\n\n\nAbstr
act\nWe resolve Manin's conjecture for all Châtelet surfaces over $\\Q$ (
surfaces given by equations of the form $x^2 + ay^2 = f(z)$) -- we establi
sh asymptotics for the number of rational points of increasing height. The
key analytic ingredient is estimating sums of Fourier coefficients of mod
ular forms along polynomial values.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/243/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martin Widmer (Royal Holloway\, University of London)
DTSTART:20250508T150000Z
DTEND:20250508T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/244
DESCRIPTION:Title: Small generators of number fields\nby Martin Widmer (Roy
al Holloway\, University of London) as part of Number Theory Web Seminar\n
\n\nAbstract\nGiven a finite field extension of the rational numbers\, how
big is the smallest height of a generator? In 1998 Wolfgang Ruppert form
ulated two precise questions on this problem. One of them is completely so
lved while the other has evolved into a conjecture. We report modest progr
ess on this conjecture and\, time permitting\, will address a question by
Ellenberg that relates small generators of number fields with upper bounds
for the $l$-torsion part of class groups. Much of this is joint work with
Shabnam Akhtari and Jeffrey Vaaler.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/244/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Zureick-Brown (Amherst College)
DTSTART:20241219T160000Z
DTEND:20241219T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/245
DESCRIPTION:Title: $\\ell$-adic Images of Galois for Elliptic Curves over Q
\nby David Zureick-Brown (Amherst College) as part of Number Theory Web Se
minar\n\n\nAbstract\nI will discuss recent joint work with Jeremy Rouse an
d Drew Sutherland on Mazur’s “Program B” — the classification of t
he possible “images of Galois” associated to an elliptic curve (equiva
lently\, classification of all rational points on certain modular curves X
H). The main result is a provisional classification of the possible images
of $l$-adic Galois representations associated to elliptic curves over $\\
Q$ and is provably complete barring the existence of unexpected rational p
oints on modular curves associated to the normalizers of non-split Cartan
subgroups and two additional genus 9 modular curves of level $49$.\n\nI wi
ll also discuss the framework and various applications (for example: a ver
y fast algorithm to rigorously compute the $l$-adic image of Galois of an
elliptic curve over $\\Q$)\, and then highlight several new ideas from the
joint work\, including techniques for computing models of modular curves
and novel arguments to determine their rational points\, a computational a
pproach that works directly with moduli and bypasses defining equations\,
and (with John Voight) a generalization of Kolyvagin’s theorem to the mo
dular curves we study.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/245/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julia Stadlmann (Stanford University)
DTSTART:20250116T160000Z
DTEND:20250116T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/246
DESCRIPTION:Title: Primes in arithmetic progressions to smooth moduli\nby J
ulia Stadlmann (Stanford University) as part of Number Theory Web Seminar\
n\n\nAbstract\nFor large $$x and coprime $a$ and $q$\, the arithmetic prog
ression $n = a \\mod q$ contains approximately $\\pi(x)/\\phi(q)$ primes u
p to $x$. For which moduli $q$ can we prove that this approximation has sm
all error terms? In this talk\, I will focus on results for smooth moduli\
, which were a key ingredient in Zhang's proof of bounded gaps between pri
mes and later improvements of Polymath. Following arguments of the Polymat
h project\, I will sketch how better equidistribution estimates for primes
in APs are linked to stronger bounds on the infimum limit of gaps between
$m$ consecutive primes. I will then show how a refinement of the $q$-van
der Corput method can be used to improve on Polymath's equidistribution es
timates and thus to obtain better bounds on short gaps between 3 or more p
rimes.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/246/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valérie Berthé (IRIF\, Université Paris Cité)
DTSTART:20250220T160000Z
DTEND:20250220T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/247
DESCRIPTION:Title: A symbolic approach to bounded remainder sets\nby Valé
rie Berthé (IRIF\, Université Paris Cité) as part of Number Theory Web
Seminar\n\n\nAbstract\nA bounded remainder set is a set with bounded (loc
al) discrepancy. We discuss dynamical and symbolic approaches to the study
of bounded remainder sets for Kronecker sequences. We focus on the case o
f Pisot parameters and show how to construct bounded remainder sets in ter
ms of multidimensional continued fractions. We also discuss convergence is
sues for multidimensional continued fractions in terms of their Lyapounov
exponents.\n\nThis is joint work with W. Steiner and J. Thuswaldner.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/247/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gergely Harcos (Alfréd Rényi Institute of Mathematics)
DTSTART:20250320T160000Z
DTEND:20250320T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/248
DESCRIPTION:Title: A new zero-free region for Rankin–Selberg $L$-functions\nby Gergely Harcos (Alfréd Rényi Institute of Mathematics) as part of
Number Theory Web Seminar\n\n\nAbstract\nI will present a new zero-free re
gion for all $\\GL(1)$-twists of $\\GL(m)×\\GL(n)$ Rankin–Selberg $L$-f
unctions. The proof is inspired by Siegel’s celebrated lower bound for D
irichlet $L$-functions at $s=1$. I will also discuss some applications. Jo
int work with Jesse Thorner.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/248/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthew de Courcy-Ireland (Stockholm University)
DTSTART:20250424T150000Z
DTEND:20250424T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/249
DESCRIPTION:Title: Cubic surfaces of Markoff type\nby Matthew de Courcy-Ire
land (Stockholm University) as part of Number Theory Web Seminar\n\n\nAbst
ract\nThe Markoff surface is a cubic surface with the special feature that
it is only quadratic in each variable separately. Exchanging the two root
s of such a quadratic produces new solutions from old\, which enabled A. A
. Markoff (senior) to find all the integer solutions. More recently\, sinc
e work of J. Bourgain\, A. Gamburd\, and P. Sarnak\, it has become possibl
e to understand how the integer solutions are related to the solutions mod
ulo primes. Given a large prime modulus\, all solutions to the congruence
can be shown to lift to integer solutions by combining their work with a c
omplementary result of W. Y. Chen\, which has recently been given a new pr
oof by D. E. Martin. The talk will survey some of these developments\, inc
luding some work in progress joint with Matthew Litman and Yuma Mizuno whe
re we adapt Martin's proof to a wider family of surfaces.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/249/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andreas Strömbergsson (Uppsala University)
DTSTART:20250327T160000Z
DTEND:20250327T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/250
DESCRIPTION:Title: An effective equidistribution result in the space of $2$-dim
ensional tori with $k$ marked points\nby Andreas Strömbergsson (Uppsa
la University) as part of Number Theory Web Seminar\n\n\nAbstract\nLet $X$
be the homogeneous space $\\Gamma\\setminus G$\, where $G$ is the semidir
ect product of $\\SL(2\,\\R)$ and a direct sum of $k$ copies of $\\R^2$\,
and where $\\Gamma$ is the subgroup of integer elements in $G$. I will pre
sent a result giving effective equidistribution of one-parameter unipotent
orbits in the space $X$. The non-effective version of this result is a sp
ecial case of Ratner's celebrated equidistribution theorem for unipotent f
lows in homogeneous dynamics\, and the particular setting which we conside
r has several known applications to problems in number theory and mathemat
ical physics. Our proof makes use of the delta method in the form develope
d by Heath-Brown. This is joint work with Anders Södergren and Pankaj Vis
he.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/250/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Péter Varjú (University of Cambridge)
DTSTART:20250306T160000Z
DTEND:20250306T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/251
DESCRIPTION:Title: Lifting in special linear groups\nby Péter Varjú (Univ
ersity of Cambridge) as part of Number Theory Web Seminar\n\n\nAbstract\nG
iven an element in $\\SL_n(\\Z/q\\Z)$\, what is the smallest element of $\
\SL_n(\\Z)$ that projects to it? We show that a lift with entries bounded
by $O(q^2 \\log q)$ always exists\, and that the exponent $2$ is best poss
ible. Time permitting we may discuss the analogous problem of finding inte
ger matrices with prescribed determinant that approximates a given matrix
with real entries. Joint work with Amitay Kamber.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/251/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shaoshi Chen (Academy of Mathematics and Systems Science\, Chinese
Academy of Sciences)
DTSTART:20250227T160000Z
DTEND:20250227T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/252
DESCRIPTION:Title: Rational-Transcendental Dichotomy Theorems on Power Series w
ith Arithmetic Restrictions\nby Shaoshi Chen (Academy of Mathematics a
nd Systems Science\, Chinese Academy of Sciences) as part of Number Theory
Web Seminar\n\n\nAbstract\nIn 1906\, Fatou proved a rational-transcendent
al dichotomy theorem on power series with integer coefficients. This theor
em has been generalized to a broader class of power series whose coefficie
nts satisfy certain arithmetic restrictions. This talk will first recall s
ome rational-transcendental dichotomy theorems in number fields and then p
resent some more recent theorems in the context of power series rings and
differential fields\, along with related conjectures and open problems. Th
is talk is based on joint work with Jason Bell\, Ehsaan Hossain\, Khoa Ngu
yen\, and Umberto Zannier.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/252/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hector Pasten (Pontificia Universidad Católica de Chile)
DTSTART:20250130T160000Z
DTEND:20250130T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/253
DESCRIPTION:Title: Effective Mordell for curves with enough automorphisms\n
by Hector Pasten (Pontificia Universidad Católica de Chile) as part of Nu
mber Theory Web Seminar\n\n\nAbstract\nThe effective Mordell conjecture as
ks for an algorithm to compute the rational points of curves of genus $g>1
$ defined over number fields. At present this is open. While there are met
hods derived from Chabauty--Coleman--Kim that in practice work extremely w
ell under some assumptions\, these methods are not known to terminate. Our
main result is an explicit and computable height bound for rational point
s of curves with "enough automorphisms"\, which gives a practical algorith
m that terminates when the relevant hypothesis is satisfied\; we will pres
ent an example. Our methods build on Arakelov geometry and sphere packing.
This is joint work with Natalia Garcia-Fritz.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/253/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stephanie Chan (IST Austria)
DTSTART:20250403T150000Z
DTEND:20250403T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/254
DESCRIPTION:Title: The Szpiro ratios of elliptic curves with prescribed torsion
\nby Stephanie Chan (IST Austria) as part of Number Theory Web Seminar
\n\n\nAbstract\nWe demonstrate that almost all elliptic curves over $\\Q$
with prescribed torsion subgroup\, when ordered by naive height\, have Szp
iro ratio arbitrarily close to the expected value. The results are achieve
d by proving that\, given any multivariate polynomial within a general cla
ss\, the absolute value of the polynomial over an expanding box is typical
ly bounded by a fixed power of its radical. The proof adapts work of Fouvr
y--Nair--Tenenbaum\, which shows that almost all elliptic curves have Szpi
ro ratio close to $1$.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/254/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mayank Pandey (Princeton University)
DTSTART:20250410T150000Z
DTEND:20250410T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/255
DESCRIPTION:Title: Squarefree numbers in short intervals and related results\nby Mayank Pandey (Princeton University) as part of Number Theory Web Se
minar\n\n\nAbstract\nWe will discuss recent work on an improved upper boun
d on the sizes of gaps between squarefree numbers. Time permitting\, we wi
ll also discuss upcoming work concerning representations of integers by te
rnary cubic linear in each variable\, in which nilsequences also arise in
a similar fashion.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/255/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Florian Karl Richter (EPFL)
DTSTART:20250313T160000Z
DTEND:20250313T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/256
DESCRIPTION:Title: Sums and products in sets of positive density\nby Floria
n Karl Richter (EPFL) as part of Number Theory Web Seminar\n\n\nAbstract\n
Hindman's conjecture states that for any finite coloring of the integers\,
there exist natural numbers $x$ and $y$ such that $x\, y\, x+y\, xy$ all
have the same color. This conjecture remains open\, with its difficulty st
emming from the challenge of controlling arithmetic structures that simult
aneously involve both addition and multiplication. In this talk\, we will
discuss how arithmetic configurations as the ones appearing in Hindman’s
conjecture are governed by the local Host-Kra uniformity norms. Our appro
ach relies on tools and ideas from multiplicative number theory. This allo
ws us to establish a density analogue of a special case of a theorem of Mo
reira and to resolve a conjecture of Moreira.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/256/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dimitris Koukoulopoulos (University of Montreal)
DTSTART:20250417T150000Z
DTEND:20250417T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/257
DESCRIPTION:Title: Erdős's integer dilation approximation problem\nby Dimi
tris Koukoulopoulos (University of Montreal) as part of Number Theory Web
Seminar\n\n\nAbstract\nLet $\\mathcal{A}\\subset\\mathbb{R}_{\\geqslant1}$
be a countable set such that $\\limsup_{x\\to\\infty}\\frac{1}{\\log x}\\
sum_{\\alpha\\in\\mathcal{A}\\cap[1\,x]}\\frac{1}{\\alpha}>0$. Erd\\H os c
onjectured in 1948 that\, for every $\\varepsilon>0$\, there exist infinit
ely many pairs $(\\alpha\, \\beta)\\in \\mathcal{A}^2$ such that $\\alpha\
\neq \\beta$ and $|n\\alpha-\\beta| <\\varepsilon$ for some positive integ
er $n$. When $\\mathcal{A}$ is a set of integers\, the conjecture follows
by work of Erd\\H os and Behrend on primitive sets of integers from the 19
30s. Moreover\, if $\\mathcal{A}$ contains ``enough elements" all of pairw
ise ratios are irrational\, then Haight proved Erdős's conjecture in 1988
. In this talk\, I will present recent joint work with Youness Lamzouri an
d Jared Duker Lichtman that solves the conjecture in full generality. A cr
itical role in the proof is played by the machinery of GCD graphs\, which
were introduced by Koukoulopoulos-Maynard in the proof of the Duffin--Scha
effer conjecture in Diophantine approximation.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/257/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Kontorovich (Rutgers University)
DTSTART:20250206T160000Z
DTEND:20250206T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/258
DESCRIPTION:Title: Spanning Trees of Graphs\nby Alex Kontorovich (Rutgers U
niversity) as part of Number Theory Web Seminar\n\n\nAbstract\nWe prove th
e exponential growth of the cardinality of the set of numbers of spanning
trees in graphs\, answering a question of Sedlacek from 1969. The proof us
es a connection with continued fractions\, Diophantine approximation\, and
advances towards Zaremba’s conjecture. This is joint work with Swee Hon
g Chan and Igor Pak.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/258/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vivian Kuperberg (ETH Zürich)
DTSTART:20250501T150000Z
DTEND:20250501T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/259
DESCRIPTION:Title: Consecutive sums of two squares in arithmetic progressions\nby Vivian Kuperberg (ETH Zürich) as part of Number Theory Web Seminar
\n\n\nAbstract\nThere are infinitely many primes whose last digit is $1$ s
uch that the next prime also ends in a $1$\, and in fact these primes have
positive density in the set of all primes. However\, it is an open proble
m to show that there are infinitely many primes ending in $1$ such that th
e next prime ends in $3$. In this talk\, we'll instead consider the sequen
ce of sums of two squares in increasing order. We'll show that there are i
nfinitely many sums of two squares ending in $1$ such that the next sum of
two squares ends in $3$\, and in fact that these sums of two squares have
positive density in the set of all sums of two squares. Joint work with N
oam Kimmel.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/259/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emmanuel Ullmo (IHES)
DTSTART:20250213T160000Z
DTEND:20250213T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/260
DESCRIPTION:Title: Bi-$\\overline{\\Q}$-Structures on Hermitian Symmetric Space
s and quadratic relations between CM periods\nby Emmanuel Ullmo (IHES)
as part of Number Theory Web Seminar\n\n\nAbstract\nWe define a natural
bi-$\\overline{\\Q}$-structure on the tangent space at a CM point on a He
rmitian locally symmetric domain. We prove that this bi-$\\overline{\\Q}$-
structure decomposes into the direct sum of 1-dimensional bi-$\\overline{\
\Q}$-subspaces\, and make this decomposition explicit for the moduli space
of abelian varieties $A_g$. We propose an "Hyperbolic Analytic Subspace"
Conjecture\, which is the analogue of Wüstholz’s Analytic Subgroup Theo
rem in this context. We show that this conjecture\, applied to $A_g$ \, im
plies that all quadratic $\\overline{\\Q}$-relations among the holomorphic
periods of CM abelian varieties arise from elementary ones. We then show
that the elementary quadratic relations between CM periods are at the hear
t of the theory: For any CM abelian variety $A$\, there exists an abelian
variety $B$ such that all the algebraic relations among CM periods on $A\\
times B$\, induced by Hodge cycles\, are generated by these elementary qua
dratic relations.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/260/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Koymans (Utrecht University)
DTSTART:20250515T150000Z
DTEND:20250515T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/261
DESCRIPTION:Title: Hilbert 10 via additive combinatorics I\nby Peter Koyman
s (Utrecht University) as part of Number Theory Web Seminar\n\n\nAbstract\
nHilbert’s tenth problem asks: given a polynomial $f$ with integer coeff
icients\, is there an algorithm to decide whether $f$ has an integer zero?
Matiyasevich\, building on earlier work of Robinson and Davis—Putnam—
Robinson\, showed that this problem is undecidable. He also asked what hap
pened if $\\Z$ is replaced with other rings of number-theoretic interest\,
for example the ring of integers $O_K$ of a number field $K$.\n\nCornelis
sen\, Poonen and Shlapentokh proved results of the following prototype: if
there exists an elliptic curve $E$ over $K$ with rank equal to $1$ and ce
rtain additional properties\, then Hilbert’s tenth problem is undecidabl
e for $O_K$. In this talk\, we will give a high-level overview of our rece
nt rank growth result on elliptic curves and show how it resolves Hilbert
’s tenth problem for every number field $K$ using a well-known reduction
argument. In part II\, Carlo Pagano will discuss further details and futu
re applications of this method. This is joint work with Carlo Pagano.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/261/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlo Pagano (Concordia University)
DTSTART:20250522T150000Z
DTEND:20250522T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/262
DESCRIPTION:Title: Hilbert 10 via additive combinatorics II\nby Carlo Pagan
o (Concordia University) as part of Number Theory Web Seminar\n\n\nAbstrac
t\nIn part I\, Peter Koymans has explained how to reduce Hilbert tenth pro
blem for finitely generated rings to a problem on rank growth of elliptic
curves\, after the work of Poonen--Shlapentokh\, and then gave a high leve
l overview of the strategy of our proof of the following rank growth theor
em. For any number field $K$ with at least $32$ real places\, there exists
an elliptic curve $E/K$ such that $rkE(K(i))=rkE(K)>0$. We will present
the proof of this Theorem and a more flexible (recently developed) version
of our method. With that in hand\, we will announce one of its novel cons
equences. This is joint work with Peter Koymans.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/262/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Will Sawin (Princeton University)
DTSTART:20250605T150000Z
DTEND:20250605T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/263
DESCRIPTION:Title: Axiomatic multiple Dirichlet series and moments of $L$-funct
ions\nby Will Sawin (Princeton University) as part of Number Theory We
b Seminar\n\n\nAbstract\nMultiple Dirichlet series are series in several c
omplex variables satisfying many functional equations. They often have app
lications to moments of Dirichlet $L$-functions. In joint work with Ian Wh
itehead we give a new construction of these series\, unifying many previou
sly constructed examples and producing new ones. Some of our examples shou
ld enable the computation of new moments of $L$-functions. Our constructio
n is in the function field case\, but it is likely possible to transfer th
ese series to number fields.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/263/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jan Hendrik Bruinier (TU Darmstadt)
DTSTART:20250626T150000Z
DTEND:20250626T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/264
DESCRIPTION:Title: Arithmetic volumes of unitary Shimura varieties\nby Jan
Hendrik Bruinier (TU Darmstadt) as part of Number Theory Web Seminar\n\n\n
Abstract\nThe geometric volume of a unitary Shimura variety can be defined
as the self-intersection number of the Hodge line bundle on it. It repres
ents an important invariant\, which can be explicitly computed in terms of
special values of Dirichlet L-functions. Analogously\, the arithmetic vol
ume is defined as the arithmetic self-intersection number of the Hodge bun
dle\, equipped with the Petersson metric\, on an integral model of the uni
tary Shimura variety. We show that such arithmetic volumes can be expresse
d in terms on logarithmic derivatives of Dirichlet $L$-functions. This is
joint work with Ben Howard.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/264/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oliver Roche-Newton (Johannes Kepler University)
DTSTART:20250612T150000Z
DTEND:20250612T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/265
DESCRIPTION:Title: Additive properties of convex sets\nby Oliver Roche-Newt
on (Johannes Kepler University) as part of Number Theory Web Seminar\n\n\n
Abstract\nA finite set $A\\subset \\mathbb R$ is said to be \\textit{conve
x} if its consecutive differences are strictly increasing. That is\, label
ling the elements of $A$ so that $a_1< a_2< \\dots a_n$\, we have that\n\\
[\n a_{i}-a_{i-1} < a_{i+1} - a_i\n\\]\nholds for all $2 \\leq i \\leq n-1
$. One expects that convex sets cannot be too additively structured\, and
there are various different problems which give different ways to quantify
this belief. Perhaps the most well-known such problem is the conjecture o
f Erdős which states that the sum set of a convex set must have cardinali
ty close to the maximum possible size $c|A|^2$.\n\nIn this talk (based on
work in progress with Thomas Bloom and Jakob Führer)\, I will discuss som
e other additive questions concerning convex sets. The central question of
the talk is the following: how many three-term arithmetic progressions ca
n a convex set have? Some partial answers to this and closely related prob
lems will be given.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/265/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Davide Lombardo (University of Pisa)
DTSTART:20250619T150000Z
DTEND:20250619T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/266
DESCRIPTION:Title: Effectivity for the integral points of certain curves of gen
us 2\nby Davide Lombardo (University of Pisa) as part of Number Theory
Web Seminar\n\n\nAbstract\nThe effective determination of the integral po
ints of an affine algebraic curve is still largely an open problem. Howeve
r\, several methods have been proposed for specific types of equations: fo
r example\, Baker's theory of linear forms handles the case of the hyperel
liptic equations $y^2=f(x)$. Geometrically\, these can be viewed as projec
tive hyperelliptic curves from which a subset consisting of one or two poi
nts "in special position" has been removed.\n\nIn joint work with Pietro C
orvaja and Umberto Zannier\, we study the simplest case for which effectiv
ity is not known in general: projective curves of genus 2 from which a sin
gle non-special point has been removed. We prove the existence of a dense
subset $T$ of the moduli space of smooth projective curves of genus 2 with
a marked point with the following property: for every $t \\in T$\, the ($
S$-)integral points on the affine curve corresponding to $t$ can be effect
ively determined over any number field. The method combines a criterion of
Bilu\, the construction of étale covers of the curve\, and the study of
torsion specialisations of sections of abelian schemes.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/266/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fabien Pazuki (University of Copenhagen)
DTSTART:20250529T150000Z
DTEND:20250529T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/267
DESCRIPTION:Title: Parallelogram inequality for abelian varieties and applicati
ons\nby Fabien Pazuki (University of Copenhagen) as part of Number The
ory Web Seminar\n\n\nAbstract\nLet $A$ be an abelian variety defined over
a number field. A theorem of Rémond states that for any two finite subgro
up schemes $G\, H$\, the Faltings height of the four isogenous abelian var
ieties $A/G\, A/H\, A/(G+H)\, A/(G\\cap H)$ are linked by an elegant inequ
ality\, which has applications in diophantine geometry. We will discuss th
e importance of the inequality\, in particular when working on explicit bo
unds on the number of torsion points in Mordell-Weil groups. The goal of t
he talk is to present an analogous inequality for abelian varieties define
d over function fields (in any characteristic). This is joint work with Ri
chard Griffon and Samuel Le Fourn.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/267/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rafael von Känel (Institute for Advanced Study\, Tsinghua Univers
ity)
DTSTART:20251030T160000Z
DTEND:20251030T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/268
DESCRIPTION:Title: Non-degenerate Diophantine equations\nby Rafael von Kän
el (Institute for Advanced Study\, Tsinghua University) as part of Number
Theory Web Seminar\n\n\nAbstract\nWe present explicit bounds for the size/
height of the solutions of Diophantine equations satisfying a certain non-
degeneracy criterion. Our result establishes in particular the effective M
ordell conjecture for large classes of (explicit) curves over the rational
numbers. In addition\, combining our explicit height bounds with Diophant
ine approximation techniques allowed us to solve the Fermat problem inside
a classical rational surface and to completely determine the set of ratio
nal points of certain explicit families of curves of genus $>1$. We discus
s these applications and we also explain the strategy of proof which combi
nes the method of Faltings (Arakelov\, Parsin\, Szpiro) with modularity an
d Masser-Wustholz isogeny estimates. Joint work with Shijie Fan.\n\nThe ta
lk will be accessible for students and several open problems will be menti
oned.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/268/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Blomer (Universität Bonn)
DTSTART:20250904T150000Z
DTEND:20250904T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/269
DESCRIPTION:Title: Perspectives on the sup-norm problem\nby Valentin Blomer
(Universität Bonn) as part of Number Theory Web Seminar\n\n\nAbstract\nT
he sup-norm problem asks for pointwise bounds of eigenfunctions on arithme
tic Riemannian manifolds\, such as the modular curve and higher dimensiona
l generalizations. The analysis of such functions offers a fascinating int
erplay of number theory\, algebra and asymptotic analysis on Lie groups wi
th applications ranging from $L$-functions and automorphic forms to Arakel
ov theory. I will survey and present various aspects of the sup-norm probl
em mostly from a number theoretic point of view.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/269/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Levent Alpöge (Harvard University)
DTSTART:20251002T150000Z
DTEND:20251002T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/270
DESCRIPTION:Title: Hilbert’s tenth problem over $ \\mathfrak{o}_K$\, $K$ a nu
mber field\nby Levent Alpöge (Harvard University) as part of Number T
heory Web Seminar\n\n\nAbstract\nI will talk about Hilbert's tenth problem
\, in its original over $\\Z$ and also its generalization to the ring of i
ntegers of a number field $K$\, the latter by now having multiple solution
s! For ours let me offer the following abstract:\n\nFor all quadratic exte
nsions $L / K$ of number fields we produce abelian varieties $A / K$ with
the same\, positive rank over $L$ and $K$. This was the last step necessar
y to solve Hilbert’s tenth problem over all (among other things) rings o
f integers of number fields. Joint work with Manjul Bhargava\, Wei Ho\, an
d Ari Shnidman.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/270/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stanley Yao Xiao (University of Northern British Columbia)
DTSTART:20250911T150000Z
DTEND:20250911T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/271
DESCRIPTION:Title: Hilbert's Tenth Problem for systems of diagonal quadratic fo
rms\, and Buchi's problem\nby Stanley Yao Xiao (University of Northern
British Columbia) as part of Number Theory Web Seminar\n\n\nAbstract\nIn
the aftermath of the negative solution to Hilbert's Tenth Problem given by
Matiyasevich (following work by Davis\, Putnam\, and Robinson)\, J.R. Buc
hi proved that solubility over the integers of arbitrary systems of diagon
al quadratic form equations is not decidable\, conditioned on the followin
g condition: for some $n_0$\, whenever $n > n_0$\, every increasing sequen
ce of $n$ positive integer squares with constant second difference equal t
o $2$ must consist of consecutive squares. In fact\, he suggested that one
can take $n_0 = 4$. We prove this assertion\, thereby showing that solubi
lity over $\\mathbb{Z}$ of arbitrary systems of diagonal quadratic form eq
uations\, the simplest non-linear systems\, is not decidable.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/271/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fedor Pakovich (Ben Gurion University of the Negev)
DTSTART:20251113T160000Z
DTEND:20251113T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/272
DESCRIPTION:Title: Holomorphic maps sharing preimages over number fields\nb
y Fedor Pakovich (Ben Gurion University of the Negev) as part of Number Th
eory Web Seminar\n\n\nAbstract\nLet \\( R \\) be a compact Riemann surface
\, \\( P: R \\to \\mathbb P^1(\\mathbb C) \\) and \\( Q: R \\to \\mathbb P
^1(\\mathbb C) \\) holomorphic maps\, and let \\( K \\) be an infinite sub
set of \\( \\mathbb P^1(\\mathbb C) \\) satisfying certain restrictions. W
e are interested in the following problem: under what conditions do the pr
eimages \\( P^{-1}(K) \\) and \\( Q^{-1}(K) \\) coincide? \nEquivalently\
, one may ask which sets \\( K \\) satisfying prescribed restrictions are
completely invariant under holomorphic correspondences. One of the very fe
w examples where a complete answer is known occurs when \\( P \\) and \\(
Q \\) are \\textit{polynomials} on \\( \\mathbb P^1(\\mathbb C) \\) and \\
( K \\) is a \\textit{compact} set. In the talk\, we present several resul
ts for the case where the restriction on \\( K \\) is that \\( K \\subset
\\mathbb P^1({\\bf k}) \\)\, with \\( {\\bf k} \\) a number field. We als
o consider the more general set-theoretic equation \\( P^{-1}(K_1) = Q^{-1
}(K_2) \\)\, where \\( K_1 \\) and \\( K_2 \\) are infinite subsets of \\(
\\mathbb P^1({\\bf k}) \\).\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/272/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mariusz Lemańczyk (Nicolaus Copernicus University)
DTSTART:20250925T150000Z
DTEND:20250925T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/273
DESCRIPTION:Title: Möbius orthogonality - ergodic viewpoint\nby Mariusz Le
mańczyk (Nicolaus Copernicus University) as part of Number Theory Web Sem
inar\n\n\nAbstract\nIn 2010\, P. Sarnak formulated the conjecture on the o
rthogonality of the Möbius function to all deterministic continuous obser
vables. I will recall connections of the Sarnak conjecture with the classi
cal Chowla conjecture from 1965\, and focus on the problem of validity of
Möbius orthogonality in different topological models of a given measure-
theoretic automorphism.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/273/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleksiy Klurman (University of Bristol)
DTSTART:20251023T150000Z
DTEND:20251023T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/274
DESCRIPTION:Title: Distribution of mixed character sums and extremal problems f
or Littlewood polynomials\nby Oleksiy Klurman (University of Bristol)
as part of Number Theory Web Seminar\n\n\nAbstract\nI will talk about the
distribution of character sums twisted by exponentials. I will discuss how
to use these results to make progress on an old problem of Mahler (constr
ucting polynomials with coefficients -1 and +1 with large Mahler measure)\
, as well as on minimizing the $L_p$ norms of well-known Turyn polynomials
. This is based on recent joint work with J. Bober and B. Shala.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/274/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amir Mohammadi (UC Berkeley)
DTSTART:20251016T150000Z
DTEND:20251016T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/275
DESCRIPTION:Title: Polynomially effective results in homogeneous dynamics and t
he Oppenheim conjecture\nby Amir Mohammadi (UC Berkeley) as part of Nu
mber Theory Web Seminar\n\n\nAbstract\nThere is a profound connection betw
een homogeneous dynamics and number theory\, particularly in the study of
Diophantine approximation. A landmark example is Margulis’s resolution o
f the Oppenheim conjecture in the mid-1980s using tools from dynamics\, fo
llowed by further strengthenings by Eskin\, Margulis\, and Mozes based on
Ratner’s seminal work. In this talk\, we will present effective results
in this direction\, with an emphasis on obtaining polynomial rates. This i
s based on joint works with Elon Lindenstrauss\, Zhiren Wang\, and Lei Yan
g.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/275/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jesse Thorner (University of Illinois\, Urbana-Champaign)
DTSTART:20251009T150000Z
DTEND:20251009T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/276
DESCRIPTION:Title: On the $\\GL(n)$ large sieve\nby Jesse Thorner (Universi
ty of Illinois\, Urbana-Champaign) as part of Number Theory Web Seminar\n\
n\nAbstract\nI will discuss a new large sieve inequality for automorphic f
orms on $\\GL(n)$ that refines and improves upon earlier works. I will hig
hlight two applications. The first is the sharpest unconditional upper bou
nd on the 2nd moment of $L$-functions (evaluated at $s = 1/2$) of cuspidal
automorphic representations in the truncated universal $\\GL(n)$ family.
The second is the removal of all unproven hypotheses in the log-free zero
density estimate for zeros of Rankin--Selberg $L$-functions in families pr
oved by Brumley\, Thorner\, and Zaman. This is joint work with Alex Pasca
di.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/276/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashvin A. Swaminathan (Harvard University)
DTSTART:20250918T150000Z
DTEND:20250918T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/277
DESCRIPTION:Title: Second moments for 2-Selmer groups and 2-class groups\nb
y Ashvin A. Swaminathan (Harvard University) as part of Number Theory Web
Seminar\n\n\nAbstract\nIn previous work with Bhargava and Shankar\, we pro
ved that the second moment of the size of the $2$-Selmer group of elliptic
curves is at most $15$. This result\, and the methods used to prove it\,
have a number of interesting applications. In this talk\, we discuss two s
uch applications: (1) bounding the second moment of the size of the $2$-cl
ass group of monogenized cubic fields\, and (2) proving that a positive pr
oportion of elliptic curves over $\\Q$ have $2$-Selmer rank $r$\, for smal
l values of $r$.\n\nThis is based on joint work with Bhargava and Shankar
and also on joint work with Bhargava\, Ho\, and Shnidman.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/277/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bryce Kerr (UNSW Canberra)
DTSTART:20251127T140000Z
DTEND:20251127T150000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/278
DESCRIPTION:Title: Poissonian pair correlation for real sequences\nby Bryce
Kerr (UNSW Canberra) as part of Number Theory Web Seminar\n\n\nAbstract\n
Poissonian pair correlation is a local statistic that captures pseudo-rand
omness in deterministic sequences. In joint work with Wang\, we provide ne
w sufficient conditions under which a real sequence exhibits the metric Po
issonian property which improves on previous results of Aistleitner\, El-B
az and Munsch.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/278/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Zywina (Cornell University)
DTSTART:20251106T160000Z
DTEND:20251106T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/279
DESCRIPTION:Title: Elliptic curves of low rank over number fields\nby David
Zywina (Cornell University) as part of Number Theory Web Seminar\n\n\nAbs
tract\nFor an elliptic curve over a number field $K$\, its set of $K$-poin
ts is a finitely generated abelian group whose rank is an important invari
ant. It is an open and difficult problem to determine which ranks occur fo
r elliptic curves over a fixed number field $K$. We will discuss recent w
ork which shows that there are infinitely many elliptic curves over $K$ of
rank $r$ for each nonnegative integer $r$ that is at most $4$. Our curv
es will be found by specializing explicit families. We will use a result
of Kai\, which generalizes work of Green\, Tao and Ziegler to number fiel
ds\, to carefully choose our curves in the families.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/279/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuri Zarhin (Pennsylvania State University)
DTSTART:20251204T160000Z
DTEND:20251204T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/280
DESCRIPTION:Title: Principally polarized abelian varieties with additional symm
etries that are not jacobians\nby Yuri Zarhin (Pennsylvania State Univ
ersity) as part of Number Theory Web Seminar\n\n\nAbstract\nWe study princ
ipally polarized complex abelian varieties $X$ of positive dimension $g$ t
hat admit a periodic automorphism of odd prime order $p$ such that its set
of fixed points is finite. By functoriality\, this automorphism acts as
a diagonalizable linear operator in the $g$-dimensional complex vector spa
ce of differentials of the first kind on $X$\; its spectrum consists of pr
imitive $p$th roots of unity. \n\nWe describe explicitly all the possible
multiplicity functions on the set of primitive $p$ roots of unity that ar
ise from canonically polarized jacobians of smooth irreducible projective
curves of genus $g$.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/280/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lillian Pierce (Duke University)
DTSTART:20260122T160000Z
DTEND:20260122T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/281
DESCRIPTION:Title: Counting points in thin sets: a big picture\nby Lillian
Pierce (Duke University) as part of Number Theory Web Seminar\n\n\nAbstrac
t\nMany problems in number theory can be framed as questions about countin
g integral solutions to a Diophantine equation\, within a box of growing s
ize. If there are very few\, or very many variables\, certain methods gain
an advantage\, but sometimes there is extra structure that can be exploit
ed as well. For example: let $f$ be a given polynomial with integer coeffi
cients in $n$ variables. How many values of $f$ are a perfect square? A pe
rfect cube? These questions arise in a variety of specific applications\,
and also in the context of a general conjecture of Serre on counting point
s in thin sets. In this talk\, we will give a broad overview of progress o
n counting points in thin sets\, including the resolution of several centr
al questions. In the context of affine thin sets of type II\, we will desc
ribe a new sieve method that is insensitive to the singularity of the unde
rlying hypersurface. This includes recent joint work with Dante Bonolis an
d Katharine Woo.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/281/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joël Ouaknine (Max Planck Institute for Software Systems)
DTSTART:20251120T160000Z
DTEND:20251120T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/282
DESCRIPTION:Title: Fragments of Hilbert’s Program\nby Joël Ouaknine (Max
Planck Institute for Software Systems) as part of Number Theory Web Semin
ar\n\n\nAbstract\nHilbert’s dream of mechanising all of mathematics was
dealt fatal blows by Gödel\, Church\, and Turing in the 1930s\, almost a
hundred years ago. Paradoxically\, assisted and automated theorem proving
have never been as popular as they are today! Motivated by algorithmic pro
blems in discrete dynamics\, nonlinear arithmetic\, and program analysis\,
we examine the decidability of various logical theories over the natural
numbers\, and discuss a range of open questions at the intersection of log
ic\, automata theory\, and number theory.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/282/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean-Louis Colliot-Thélène (CNRS/Université Paris-Saclay)
DTSTART:20251211T160000Z
DTEND:20251211T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/283
DESCRIPTION:Title: On the stable rationality of certain real threefolds\nby
Jean-Louis Colliot-Thélène (CNRS/Université Paris-Saclay) as part of N
umber Theory Web Seminar\n\n\nAbstract\nOver the reals\, we investigate s
table rationality of smooth projective threefolds of the following types
: intersections of two quadrics in $5$-dimensional projective space and t
hreefolds with a fibration into quadrics surfaces over the projective line
\, under the (necessary) condition that the set of real points is connecte
d.\n\nOver the field $R$ of real Puiseux series (a real closed field)\, we
construct varieties of each of these types which are not stably rational
but for which the space $X(R)$ of $R$-points is semi-algebraically connect
ed. The question of constructing such examples over the field of real numb
ers remains open.\n\nWe also consider specific quadric bundles over the r
eals from the point of view of decomposition of the diagonal (a property w
eaker than stable rationality). A specific case is given by affine equati
ons $x^2+y^2+z^2=u.p(u)$ over the reals\, with $p(u)$ a positive polynomia
l of degree $2$. If the j-invariant of the real elliptic curve $z^2=u.p(u
)$ is nonnegative\, then there is a decomposition of the diagonal.\n\nJoin
t works with Alena Pirutka and with Federico Scavia.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/283/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Loughran (University of Bath)
DTSTART:20260115T160000Z
DTEND:20260115T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/284
DESCRIPTION:Title: Arithmetic statistics via algebraic stacks\nby Daniel Lo
ughran (University of Bath) as part of Number Theory Web Seminar\n\n\nAbst
ract\nIn this talk I will explain some recent interesting applications of
the theory of algebraic stacks to the area of arithmetic statistics\, name
ly to counting number fields (Malle's conjecture) and the distribution of
class groups (the Cohen-Lenstra heuristics). In particular I will explain
a recent conjecture of myself with Tim Santens on the leading constant in
Malle's conjecture. No knowledge of algebraic stacks required.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/284/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alin Bostan (INRIA\, Sorbonne University)
DTSTART:20251218T160000Z
DTEND:20251218T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/285
DESCRIPTION:Title: On Deciding Transcendence of Power Series\nby Alin Bosta
n (INRIA\, Sorbonne University) as part of Number Theory Web Seminar\n\n\n
Abstract\nA power series is said to be D-finite (“differentially finite
”) if it satisfies a linear differential equation with polynomial coeffi
cients. D-finite power series are ubiquitous in combinatorics\, number the
ory and mathematical physics. In his seminal article on D-finite functions
[S1]\, Richard P. Stanley asked for “an algorithm suitable for computer
implementation” to decide whether a given D-finite power series is alge
braic or transcendental. Although Stanley insisted on the practical aspect
of the targeted algorithm\, at the time he formulated the problem it was
unknown whether the task of recognizing algebraicity of D-finite functions
is decidable even in theory. I will first report on such a decidability r
esult. The corresponding algorithm has too high a complexity to be useful
in practice. This is because it relies on the costly algorithm from [S2]\,
which involves\, among other things\, factoring linear differential opera
tors\, solving huge polynomial systems and solving Abel’s problem. I wil
l then present an answer to Stanley’s question based on “minimization
” of linear differential equations\, and illustrate it through examples
coming from combinatorics and number theory. (Work in collaboration with B
runo Salvy and Michael F. Singer.)\n\n[S1] R. P. Stanley\, "Differentiably
finite power series". European J. Combin. 1 (1980)\, no. 2\, 175–188.\n[S2] M. F. Singer\, "Algebraic solutions of nth order linear different
ial equations". Proc. Queen’s Number Theory Conf. 1979\, Queen's Papers
in Pure and Appl. Math.\, 54 (1980)\, 379–420.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/285/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Floris Vermeulen (University of Münster)
DTSTART:20260319T160000Z
DTEND:20260319T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/286
DESCRIPTION:Title: Serre's thin set conjecture\nby Floris Vermeulen (Univer
sity of Münster) as part of Number Theory Web Seminar\n\n\nAbstract\nIn t
he eighties\, Serre conjectured upper bounds for counting rational points
on thin sets in projective space. Thin sets of type I come from subvarieti
es\, while thin sets of type II come from dominant finite covers of projec
tive space. I will give an introduction to thin sets and give an overview
of counting rational points on thin sets. I will then discuss recent work
on type II thin sets via the determinant method. This is based on joint wo
rk with Tijs Buggenhout\, Raf Cluckers\, Per Salberger and Tim Santens.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/286/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joni Teräväinen (University of Cambridge)
DTSTART:20260226T160000Z
DTEND:20260226T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/287
DESCRIPTION:Title: Prime factorisations of consecutive integers\nby Joni Te
räväinen (University of Cambridge) as part of Number Theory Web Seminar\
n\n\nAbstract\nWe will discuss recent progress on several conjectures of E
rdős and collaborators concerning the arithmetic function ω(n)\, includi
ng a conjecture of Erdős and Straus on long strings of integers with few
prime factors\, Erdős's irrationality conjecture for a series involving
ω(n)\, and the Erdős–Pomerance–Sárközy conjecture on the frequency
of solutions to ω(n)=ω(n+1). A common theme is the interplay between pr
obabilistic methods\, sieves\, and quantitative correlation estimates for
multiplicative functions. I will outline how these tools allow us to resol
ve the first two conjectures and to verify the third for almost all values
of x. This is based on joint work with Terence Tao.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/287/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Newton (University of Oxford)
DTSTART:20260219T160000Z
DTEND:20260219T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/288
DESCRIPTION:Title: The Ramanujan conjecture for Bianchi modular forms and the D
work family\nby James Newton (University of Oxford) as part of Number
Theory Web Seminar\n\n\nAbstract\nI'll talk about joint work with George B
oxer\, Frank Calegari\, Toby Gee and Jack Thorne from a couple of years ag
o\, in which we prove some well-known conjectures (Ramanujan and Sato--Tat
e) for Bianchi modular forms. (Shortly after our work appeared\, Kojiro Ma
tsumoto proved more general results of this type.) In this talk\, one ingr
edient I would like to say something about is an application to the Dwork
family of projective hypersurfaces of a result of Drinfeld and Kedlaya on
p-adic valuations of Frobenius eigenvalues.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/288/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ari Shnidman (Temple University)
DTSTART:20260430T150000Z
DTEND:20260430T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/289
DESCRIPTION:by Ari Shnidman (Temple University) as part of Number Theory W
eb Seminar\n\nAbstract: TBA\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/289/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Junxian Li (UC Davis)
DTSTART:20260129T160000Z
DTEND:20260129T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/290
DESCRIPTION:Title: Shifted convolution problems and L-functions\nby Junxian
Li (UC Davis) as part of Number Theory Web Seminar\n\n\nAbstract\nA shift
ed convolution problem seeks an asymptotic formula for sums involving the
product of two arithmetic functions whose arguments differ by an additive
shift. Such problems arise naturally in the study of correlations of arith
metic functions and are also closely connected to moments of $L$-functions
. In joint work with Valentin Blomer\, we investigate the shifted convolut
ion of Fourier coefficients of $GL(3)$ cusp forms and the divisor function
\, resolving the final remaining case of shifted convolution problems for
$GL(3)\\times GL(2)$. The proof relies on two intertwined applications of
different types of delta symbol methods. As an application\, we obtain an
asymptotic formula for central values of $L$-functions associated with a $
GL(3)$ automorphic form twisted by Dirichlet characters of modulus $q\\leq
Q$.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/290/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Smith (Northwestern University)
DTSTART:20260212T160000Z
DTEND:20260212T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/291
DESCRIPTION:Title: Diophantine approximation for hypersurfaces\nby Alexande
r Smith (Northwestern University) as part of Number Theory Web Seminar\n\n
\nAbstract\nAmong the nondegenerate $C^4$ hypersurfaces\, we characterize
the rational quadrics as the hypersurfaces that are the least well approxi
mated by rational points. For all other hypersurfaces\, we give a heuristi
cally sharp lower bound for the number of rational points near them\, impr
oving the sensitivity of prior results of Beresnevich and Huang. Our metho
ds are dynamical\, involving the application of Ratner's theorems for unip
otent orbits\, and we will show how our work relates to the dynamical reso
lution of the Oppenheim conjecture by Margulis.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/291/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Shusterman (Weizmann Institute of Science)
DTSTART:20260402T150000Z
DTEND:20260402T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/292
DESCRIPTION:Title: Sums of Root Numbers of Artin $L$-functions\nby Mark Shu
sterman (Weizmann Institute of Science) as part of Number Theory Web Semin
ar\n\n\nAbstract\nNormalized Gauss sums lie on the unit circle\, and for q
uadratic characters can be pinned down precisely. For characters of higher
(fixed) order the location on the unit circle is more random\, as the mod
ulus varies.\n\nCorresponding to a Dirichlet character is a one-dimensiona
l representation of the absolute Galois group the root number of whose $L$
-function is our Gauss sum.\n\nWe consider the distribution of root number
s of Artin $L$-functions of higher-dimensional complex representations\, f
ocusing on the function field case and its connection with homological sta
bility.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/292/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brad Rodgers (Queen's University)
DTSTART:20260423T150000Z
DTEND:20260423T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/293
DESCRIPTION:by Brad Rodgers (Queen's University) as part of Number Theory
Web Seminar\n\nAbstract: TBA\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/293/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carl Pomerance (Dartmouth College)
DTSTART:20260312T160000Z
DTEND:20260312T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/294
DESCRIPTION:Title: The Erdős--Straus conjecture\nby Carl Pomerance (Dartmo
uth College) as part of Number Theory Web Seminar\n\n\nAbstract\nIn 1948 E
rdős and Straus conjectured that for every\ninteger $n>1$\, the fraction
$4/n$ is equal to $1/a + 1/b + 1/c$\nfor some positive integers $a\, b\, c
$. Still unsolved after\nnearly 80 years\, this curious conjecture has be
en studied\nby Sierpinksi\, Schinzel\, Mordell\, Vaughan\, Elsholtz \\& Ta
o\,\nand many others. This talk will review what is known and\ndiscuss so
me new results. (Joint work with Andreas Weingartner.)\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/294/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Kalmynin (HSE University)
DTSTART:20260305T160000Z
DTEND:20260305T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/295
DESCRIPTION:Title: Additive irreducibility of multiplicative subgroups\nby
Alexander Kalmynin (HSE University) as part of Number Theory Web Seminar\n
\n\nAbstract\nA central theme in additive combinatorics is the heuristic t
hat sets which are highly structured multiplicatively should not exhibit r
ich additive structure. One notable manifestation of this principle is a c
onjecture due to Sárközy\, which states that for a sufficiently large pr
ime $p$\, the set $R_p$ of non-zero quadratic residues modulo $p$ cannot b
e expressed as a sumset $A+B$ with $\\min(|A|\,|B|)>1$. The difference var
iant of this conjecture was formulated by Lev and Sonn: if $R_p$ is the se
t of non-zero differences for some set $A$ and every $r$ in $R_p$ is uniqu
ely represented as $a_1-a_2$ with $a_1\,a_2$ from $A$\, then $p=5$ or $13$
. In this talk we present full resolutions of these conjectures. The proof
bulids on and extend a variant of Stepanov's method developed by Hanson a
nd Petridis in 2019. Further\, our method shows that if a proper subgroup
G in a prime finite field admits a non-trivial decomposition $G=A+B$\, the
n $|A|=|B|=|G|^{1/2}$.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/295/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jori Merikoski (University of Helsinki)
DTSTART:20260409T150000Z
DTEND:20260409T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/296
DESCRIPTION:Title: On the greatest prime factor and uniform equidistribution of
quadratic polynomials\nby Jori Merikoski (University of Helsinki) as
part of Number Theory Web Seminar\n\n\nAbstract\nWe show that the greatest
prime factor of $n^2+h$ is at least $n^{1.312}$ infinitely often. This pr
ovides an unconditional proof for the exponent previously known under the
Selberg eigenvalue conjecture. Furthermore\, we get the same exponent unif
ormly in $h \\leq n$ under a natural hypothesis on real characters. Same u
niformity in $h$ is obtained for the equidistribution of the roots of quad
ratic congruences modulo primes\, extending Duke\, Friedlander\, and Iwani
ec who famously proved equidistribution for a fixed polynomial. The talk i
s based on joint work with L. Grimmelt. Instead of the widely used sums of
Kloosterman sums methods\, we develop a new approach based on weighted av
erages of $SL(2\,\\R)$ automorphic kernel.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/296/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sary Drappeau (Université Clermont-Auvergne)
DTSTART:20260507T150000Z
DTEND:20260507T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/297
DESCRIPTION:by Sary Drappeau (Université Clermont-Auvergne) as part of Nu
mber Theory Web Seminar\n\nAbstract: TBA\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/297/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michel Waldschmidt (Sorbonne University)
DTSTART:20260416T150000Z
DTEND:20260416T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/298
DESCRIPTION:Title: Multicyclotomy\nby Michel Waldschmidt (Sorbonne Universi
ty) as part of Number Theory Web Seminar\n\n\nAbstract\nIn a joint work in
progress with Étienne Fouvry we define a multicyclotomic polynomial as a
monic polynomial in one variable that is a product of distinct cyclotomic
polynomials. Hence a polynomial with integer coefficients is multicycloto
mic if and only if it is monic with all its roots simple and roots of unit
y. It is equivalent to say that it is a divisor of a polynomial of the for
m $T^n-1$\, or that it is separable with Mahler's measure $1$. A multicycl
otomic form is a binary form obtained by homogenizing a multicyclotomic po
lynomial. We extend to this new setting some of the results known for cycl
otomic polynomials and forms.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/298/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bianca Viray (University of Washington)
DTSTART:20260521T150000Z
DTEND:20260521T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/299
DESCRIPTION:by Bianca Viray (University of Washington) as part of Number T
heory Web Seminar\n\nAbstract: TBA\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/299/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Chow (University of Warwick)
DTSTART:20260326T160000Z
DTEND:20260326T170000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/300
DESCRIPTION:Title: Multiplicative diophantine approximation on manifolds\nb
y Sam Chow (University of Warwick) as part of Number Theory Web Seminar\n\
n\nAbstract\nWe establish the convergence theory of multiplicative diophan
tine approximation on manifolds\, in both the curved and flat settings. Th
e problem lies at the intersection of three topics: (i) Littlewood’s con
jecture\, (ii) metric diophantine approximation\, and (iii) rational point
s near manifolds. This is joint work with Rajula Srivastava\, Niclas Techn
au\, and Han Yu.\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/300/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Landesman (Harvard University)
DTSTART:20260514T150000Z
DTEND:20260514T160000Z
DTSTAMP:20260409T222208Z
UID:NTWebSeminar/301
DESCRIPTION:by Aaron Landesman (Harvard University) as part of Number Theo
ry Web Seminar\n\nAbstract: TBA\n
LOCATION:https://master.researchseminars.org/talk/NTWebSeminar/301/
END:VEVENT
END:VCALENDAR