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BEGIN:VEVENT
SUMMARY:Shalom Eliahou (Universite du Littoral Cote d'Opale)
DTSTART;VALUE=DATE-TIME:20200710T150000Z
DTEND;VALUE=DATE-TIME:20200710T160000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/1
DESCRIPTION:Title: Iterated sumsets and Hilbert functions\n
by Shalom Eliahou (Universite du Littoral Cote d'Opale) as part of New Yor
k Number Theory Seminar\n\n\nAbstract\nLet $A\,B \\subset \\Z$. Denote $A+
B=\\{a+b \\mid a \\in A\, b \\in B\\}$\, the \\emph{sumset} of $A\,B$. For
$A=B$\, denote $2A=A+A$. More generally\, for $h \\ge 2$\, denote $hA=A+(
h-1)A$\, the $h$-fold \\emph{iterated sumset} of $A$. If $A$ is finite\, h
ow does the sequence $|hA|$ behave as $h$ grows? This is a typical problem
in additive combinatorics. In this talk\, we focus on the following speci
fic question: if $|hA|$ is known\, what can one say about $|(h-1)A|$ and $
|(h+1)A|$? It is known that $$|(h-1)A| \\ge |hA|^{(h-1)/h}\,$$ a consequen
ce of Pl\\"unnecke's inequality derived from graph theory. Here we propose
a new approach\, by modeling the sequence $|hA|$ with the Hilbert functio
n of a suitable standard graded algebra $R(A)$. We then apply Macaulay's 1
927 theorem on the growth of Hilbert functions. This allows us to recover
and strengthen Pl\\"unnecke's estimate on $|(h-1)A|$. This is joint work
with Eshita Mazumdar.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20200903T190000Z
DTEND;VALUE=DATE-TIME:20200903T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/2
DESCRIPTION:Title: Sums of finite sets of integers\, II\nby
Mel Nathanson (CUNY) as part of New York Number Theory Seminar\n\nAbstrac
t: TBA\n
LOCATION:https://master.researchseminars.org/talk/New_York_Number_Theory_S
eminar/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20200910T190000Z
DTEND;VALUE=DATE-TIME:20200910T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/3
DESCRIPTION:Title: Chromatic sumsets\nby Mel Nathanson (CUN
Y) as part of New York Number Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://master.researchseminars.org/talk/New_York_Number_Theory_S
eminar/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20200917T190000Z
DTEND;VALUE=DATE-TIME:20200917T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/4
DESCRIPTION:Title: A curious convergent series of integers with
missing digits\nby Mel Nathanson (CUNY) as part of New York Number Th
eory Seminar\n\n\nAbstract\nBy a classical theorem of Kempner\, the sum of
the reciprocals of integers with missing digits converges. This result i
s extended to a much larger family of ``missing digits'' sets of positive
integers with convergent harmonic series. Related sets with divergent har
monic series are also constructed.\n
LOCATION:https://master.researchseminars.org/talk/New_York_Number_Theory_S
eminar/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Harald Helfgott (Gottigen)
DTSTART;VALUE=DATE-TIME:20200924T190000Z
DTEND;VALUE=DATE-TIME:20200924T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/5
DESCRIPTION:Title: Expansion in a prime divisibility graph\
nby Harald Helfgott (Gottigen) as part of New York Number Theory Seminar\n
\n\nAbstract\n(Joint with M. Radziwill.)\nLet $\\mathbf{N} = \\mathbb{Z} \
\cap (N\, 2N]$ and $\\mathbf{P} \\subset [1\,H]$ a set of primes\n with
$H \\leq \\exp(\\sqrt{\\log N}/2)$. Given any subset $\\mathcal{X} \\subs
et \\mathbf{N}$\,\ndefine the linear operator\n $$\n (A_{|\\mathcal{X}} f
)(n) = \\sum_{\\substack{p \\in \\mathbf{P} : p | n \\\\ n\, n \\pm p \\in
\\mathcal{X}}} f(n \\pm p) - \\sum_{\\substack{p \\in \\mathbf{P} \\\\ n\
, n \\pm p \\in \\mathcal{X}}} \\frac{f(n \\pm p)}{p}\n $$\non functions
$f:\\mathbf{N}\\to \\mathbb{C}$. Let $\\mathcal{L} = \\sum_{p \\in \\mathb
f{P}} \\frac{1}{p}$.\n\nWe prove that\, for any $C > 0$\, there exists a s
ubset $\\mathcal{X} \\subset \\mathbf{N}$ of density $1 - O(e^{-C \\mathca
l{L}})$ in $\\mathbf{N}$ such that\n$A_{|\\mathcal{X}}$ has a strong expan
der property:\nevery eigenvalue of $A_{|\\mathcal{X}}$ is $O(\\sqrt{\\math
cal{L}})$.\nIt follows immediately that\, for any bounded\n $f\,g:\\math
bf{N}\\to \\mathbb{C}$\,\n \\begin{equation}\\label{eq:bamidyar}\n \\f
rac{1}{N \\mathcal{L}} \\Big|\n \\sum_{\\substack{n \\in \\mathbf{N} \\\\
p \\in \\mathbf{P} : p | n}} f(n) \\overline{g(n\\pm p)} -\n \\sum_{\\su
bstack{n \\in \\mathbf{N} \\\\ p \\in \\mathbf{P}}} \\frac{f(n)\\overline{
g(n\\pm p)}}{p} \\Big| =\n O\\Big(\\frac{1}{\\sqrt{\\mathcal{L}}}\\Big).\
n \\end{equation}\n This bound is sharp up to constant factors.\n\n Spe
cializing the above bound to $f(n) = g(n) = \\lambda(n)$ with $\\lambda(n)
$ the Liouville function\, and using a result in (Matom\\"aki-Radziwi\\l\\
l-Tao\, 2015)\,\n we obtain\n \\begin{equation}\\label{eq:cruciator}\n
\\frac{1}{\\log x} \\sum_{n\\leq x} \\frac{\\lambda(n) \\lambda(n+1)}{n}
=\n O\\left(\\frac{1}{\\sqrt{\\log \\log x}}\\right)\,\n \\end{equa
tion}\n improving on a result of Tao's. Tao's result relied on a differen
t\n approach (entropy decrement)\, requiring $H\\leq (\\log N)^{o(1)}$\n
and leading to weaker bounds.\n\n We also prove the stronger statement\n
that Chowla's conjecture is true at almost all scales\n with an error t
erm as in (\\ref{eq:cruciator})\,\n improving on a result by Tao and Tera
v\\"ainen.\n
LOCATION:https://master.researchseminars.org/talk/New_York_Number_Theory_S
eminar/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20201001T190000Z
DTEND;VALUE=DATE-TIME:20201001T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/6
DESCRIPTION:Title: Convergent and divergent series of integers
with missing digits\nby Mel Nathanson (CUNY) as part of New York Numbe
r Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://master.researchseminars.org/talk/New_York_Number_Theory_S
eminar/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brian Hopkins (Saint Peter's University)
DTSTART;VALUE=DATE-TIME:20201008T190000Z
DTEND;VALUE=DATE-TIME:20201008T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/8
DESCRIPTION:Title: Rank\, crank\, and mex: New connections betw
een partition statistics\nby Brian Hopkins (Saint Peter's University)
as part of New York Number Theory Seminar\n\n\nAbstract\nAbout 100 years a
go\, Ramanujan proved certain patterns in the counts of integer partitions
\, but not in a way that fully ``explained'' them. A young Freeman Dyson
wrote in a somewhat cheeky 1944 article that a new notion he called the r
ank of a partition explained some of the patterns of partition counts---wi
thout proving it---and that something called the crank should explain the
rest---without defining crank! Everything he proposed was eventually prov
en by others to be correct. The new part of the story is recent work of t
he speaker and James Sellers that explains crank\, whose definition is som
ewhat tricky\, in terms of the minimal excluded part (``mex'') of integer
partitions. This allows us to improve and simplify a recent result in the
Ramanujan Journal.\n
LOCATION:https://master.researchseminars.org/talk/New_York_Number_Theory_S
eminar/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20201015T190000Z
DTEND;VALUE=DATE-TIME:20201015T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/9
DESCRIPTION:Title: Dirichlet series of integers with missing di
gits\nby Mel Nathanson (CUNY) as part of New York Number Theory Semina
r\n\n\nAbstract\nFor certain sequences $A$ of positive integers with missi
ng $g$-adic digits\, the Dirichlet series $F_A(s) = \\sum_{a\\in A} a^{-s}
$ has abscissa of convergence $\\sigma_c < 1$. The number $\\sigma_c$ is
computed. This generalizes and strengthens a classical theorem of Kempne
r on the convergence of the sum of the reciprocals of a sequence of intege
rs with missing decimal digits.\n
LOCATION:https://master.researchseminars.org/talk/New_York_Number_Theory_S
eminar/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthias Beck (San Francisco State University)
DTSTART;VALUE=DATE-TIME:20201029T190000Z
DTEND;VALUE=DATE-TIME:20201029T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/10
DESCRIPTION:Title: The arithmetic of Coxeter permutahedra\
nby Matthias Beck (San Francisco State University) as part of New York Num
ber Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://master.researchseminars.org/talk/New_York_Number_Theory_S
eminar/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthias Beck (San Francisco State University)
DTSTART;VALUE=DATE-TIME:20201029T190000Z
DTEND;VALUE=DATE-TIME:20201029T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/11
DESCRIPTION:Title: The arithmetic of Coxeter permutahedra\
nby Matthias Beck (San Francisco State University) as part of New York Num
ber Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://master.researchseminars.org/talk/New_York_Number_Theory_S
eminar/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthias Beck (San Francisco State University)
DTSTART;VALUE=DATE-TIME:20201029T190000Z
DTEND;VALUE=DATE-TIME:20201029T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/12
DESCRIPTION:Title: The arithmetic of Coxeter permutahedra\
nby Matthias Beck (San Francisco State University) as part of New York Num
ber Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://master.researchseminars.org/talk/New_York_Number_Theory_S
eminar/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthias Beck (San Francisco State University)
DTSTART;VALUE=DATE-TIME:20201029T190000Z
DTEND;VALUE=DATE-TIME:20201029T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/13
DESCRIPTION:Title: The arithmetic of Coxeter permutahedra\
nby Matthias Beck (San Francisco State University) as part of New York Num
ber Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://master.researchseminars.org/talk/New_York_Number_Theory_S
eminar/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emma Bailey (CUNY Graduate Center)
DTSTART;VALUE=DATE-TIME:20201112T200000Z
DTEND;VALUE=DATE-TIME:20201112T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/14
DESCRIPTION:Title: L-functions and random matrix theory\nb
y Emma Bailey (CUNY Graduate Center) as part of New York Number Theory Sem
inar\n\n\nAbstract\nI will review the (conjectured but well evidenced) con
nection between families of $L$-functions and characteristic polynomials o
f random matrices. The canonical example connects the Riemann zeta functio
n with unitary matrices. I will then explain some recent results pertainin
g to various moments of interest (both of characteristic polynomials and o
f $L$-functions). Our work has further connections to log-correlated fiel
ds and combinatorics. This is joint work with Jon Keating and Theo Assio
tis.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Iosevich (University of Rochester)
DTSTART;VALUE=DATE-TIME:20201105T200000Z
DTEND;VALUE=DATE-TIME:20201105T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/15
DESCRIPTION:Title: Discrete energy and applications to Erdos t
ype problems\nby Alex Iosevich (University of Rochester) as part of Ne
w York Number Theory Seminar\n\n\nAbstract\nWe are going to survey a simpl
e conversion mechanism that allows one to deduce certain quantitative disc
rete results from their continuous analogs.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNYMinimal bases in additive number theory)
DTSTART;VALUE=DATE-TIME:20201119T200000Z
DTEND;VALUE=DATE-TIME:20201119T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/16
DESCRIPTION:Title: Minimal bases in additive number theory
\nby Mel Nathanson (CUNYMinimal bases in additive number theory) as part o
f New York Number Theory Seminar\n\n\nAbstract\nThe set $A$ of nonnegative
integers is an \\emph{asymptotic basis of order $h$} if every \n sufficie
ntly large integer can be represented as the sum of $h$ elements of $A$.
\n An asymptotic basis of order $h$ is \\emph{minimal} if no proper subset
of $A$ \n is an asymptotic basis of order $h$. Minimal asymptotic bases
are extremal objects \n in additive number theory\, and related to the con
jecture of Erd\\H os and Tur\\' an that \n the representation function of
an asymptotic basis must be unbounded. \n This talk describes the constru
ction of a new class of minimal asymptotic bases.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20201203T200000Z
DTEND;VALUE=DATE-TIME:20201203T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/17
DESCRIPTION:Title: Sidon sets and perturbations\nby Mel Na
thanson (CUNY) as part of New York Number Theory Seminar\n\n\nAbstract\nA
subset $A$ of an additive abelian group is an $h$-Sidon set if every eleme
nt in the $h$-fold sumset \n$hA$ has a unique representation as the sum of
$h$ not necessarily distinct elements of $A$. \nLet $\\mathbf{F}$ be a
field of characteristic 0 with a nontrivial absolute value\, \nand let $A
= \\{a_i :i \\in \\mathbf{N} \\}$ and $B = \\{b_i :i \\in \\mathbf{N} \\}$
be subsets of $\\mathbf{F}$.\nLet $\\varepsilon = \\{ \\varepsilon_i:i
\\in \\mathbf{N} \\}$\, where $\\varepsilon_i > 0$ for all $i \\in \\math
bf{N}$.\nThe set $B$ is an $\\varepsilon$-perturbation of $A$ \nif $|b_i-
a_i| < \\varepsilon_i$ for all $i \\in \\mathbf{N}$.\nIt is proved that\,
for every $\\varepsilon = \\{ \\varepsilon_i:i \\in \\mathbf{N} \\}$ wi
th $\\varepsilon_i > 0$\, \nevery set $A = \\{a_i :i \\in \\mathbf{N} \\
}$ has an $\\varepsilon$-perturbation $B$ \nthat is an $h$-Sidon set. Th
is result extends to sets of vectors \nin $\\mbF^n$.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20201210T200000Z
DTEND;VALUE=DATE-TIME:20201210T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/18
DESCRIPTION:Title: Multiplicative representations of integers
and Ramsey's theorem\nby Mel Nathanson (CUNY) as part of New York Numb
er Theory Seminar\n\n\nAbstract\nLet $\\mathcal{B} = (B_1\,\\ldots\, B_h)$
be an $h$-tuple of sets of positive integers. \nLet $g_{\\mathcal{B}}(n)
$ count the number of multiplicative representations of $n$ \nin the form
$n = b_1\\cdots b_h$\, \nwhere $b_i \\in B_i$ for all $i \\in \\{1\,\\ldot
s\, h\\}$. \nIt is proved that $\\liminf_{n\\rightarrow \\infty} g_{\\mat
hcal{B}}(n) \\geq 2$ \nimplies $\\limsup_{n\\rightarrow \\infty} g_{\\math
cal{B}}(n) = \\infty$.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arindam Biswas (Technion\, Israel)
DTSTART;VALUE=DATE-TIME:20201217T200000Z
DTEND;VALUE=DATE-TIME:20201217T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/19
DESCRIPTION:Title: Direct and inverse problems related to mini
mal complements\nby Arindam Biswas (Technion\, Israel) as part of New
York Number Theory Seminar\n\n\nAbstract\nMinimal complements of subsets o
f groups have been popular objects of study in recent times. The notion wa
s introduced by Nathanson in 2011. The past few years have seen a flurry
of activities focussing on the existence and nonexistence of minimal com
plements. In this talk\, we shall speak about the direct and the inverse p
roblems elated to minimal complements and discuss some of the recent resu
lts addressing some of these problems.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nathan Kaplan (University of California\, Irvine)
DTSTART;VALUE=DATE-TIME:20210128T200000Z
DTEND;VALUE=DATE-TIME:20210128T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/20
DESCRIPTION:Title: Counting subrings of Z^n\nby Nathan Kap
lan (University of California\, Irvine) as part of New York Number Theory
Seminar\n\n\nAbstract\nHow many subgroups of $\\mathbb{Z}^n$ have index at
most $X$? How many of these subgroups are also subrings? We can give an
asymptotic answer to the first question by computing the ‘subgroup zeta
function’ of $\\mathbb{Z}^n$. For the second question\, we only know a
n asymptotic answer for small $n$ because the ‘subring zeta function’
of $\\mathbb{Z}^n$ is much harder to compute. It is not difficult to show
that it is enough to understand the number of subrings of prime power ind
ex. Let $f_n(p^e)$ be the number of subrings of $\\mathbb{Z}^n$ with inde
x $p^e$. When $n$ and $e$ are fixed\, how does $f_n(p^e)$ vary as a funct
ion of p? We will discuss the quotient $\\mathbb{Z}^n/L$ where $L$ is a `
random’ subgroup or subring of $\\mathbb{Z}^n$. We will also see connec
tions to counting orders in number fields.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yoshiharu Kohayakawa (University of Sao Paulo\, Brazil)
DTSTART;VALUE=DATE-TIME:20200204T200000Z
DTEND;VALUE=DATE-TIME:20200204T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/21
DESCRIPTION:Title: The number of Sidon sets and an application
to an extremal problem for random sets of integers\nby Yoshiharu Koha
yakawa (University of Sao Paulo\, Brazil) as part of New York Number Theor
y Seminar\n\nAbstract: TBA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yoshiharu Kohayakawa (University of Sao Paulo\, Brazil)
DTSTART;VALUE=DATE-TIME:20200204T200000Z
DTEND;VALUE=DATE-TIME:20200204T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/22
DESCRIPTION:Title: The number of Sidon sets and an application
to an extremal problem for random sets of integers\nby Yoshiharu Koha
yakawa (University of Sao Paulo\, Brazil) as part of New York Number Theor
y Seminar\n\nAbstract: TBA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yoshiharu Kohayakawa (University of Sao Paulo\, Brazil)
DTSTART;VALUE=DATE-TIME:20200204T200000Z
DTEND;VALUE=DATE-TIME:20200204T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/23
DESCRIPTION:Title: The number of Sidon sets and an application
to an extremal problem for random sets of integers\nby Yoshiharu Koha
yakawa (University of Sao Paulo\, Brazil) as part of New York Number Theor
y Seminar\n\nAbstract: TBA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yoshiharu Kohayakawa (University of Sao Paulo\, Brazil)
DTSTART;VALUE=DATE-TIME:20210204T200000Z
DTEND;VALUE=DATE-TIME:20210204T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/24
DESCRIPTION:Title: The number of Sidon sets and an extremal pr
oblem for random sets of integers\nby Yoshiharu Kohayakawa (University
of Sao Paulo\, Brazil) as part of New York Number Theory Seminar\n\n\nAbs
tract\nA set of integers is a Sidon set if the pairwise sums of its elemen
ts are all distinct. We discuss the number of Sidon sets contained in $[n]
=\\{1\,\\dots\,n\\}$. As an application\, we investigate random sets of i
ntegers $R\\subset[n]$ of a given\ncardinality $m=m(n)$ and study $F(R)$\,
the typical maximal cardinality of a Sidon set contained in $R$. The beh
aviour of $F(R)$ as $m=m(n)$ varies is somewhat unexpected\, presenting t
wo points of ``phase transition.'' We shall also briefly discuss the case
in which the random set $R$ is\nan infinite random subset of the set of na
tural numbers\, according to\na natural model\; that is\, we shall discuss
infinite Sidon sets\ncontained in certain infinite random sets of integer
s. Finally\, we shall mention extensions to $B_h$-sets. Joint work with D
. Dellamonica Jr.\, S. J. Lee\, C. G. Moreira\, V. R\\"odl\, and W. Samot
ij.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20210211T200000Z
DTEND;VALUE=DATE-TIME:20210211T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/25
DESCRIPTION:Title: Sidon sets for linear forms\nby Mel Nat
hanson (CUNY) as part of New York Number Theory Seminar\n\n\nAbstract\nLet
$\\varphi(x_1\,\\ldots\, x_h) = c_1 x_1 + \\cdots + c_h x_h $ be a linea
r form \nwith coefficients in a field $\\mathbf{F}$\, and let $V$ be a vec
tor space over $\\mathbf{F}$. \nA nonempty subset $A$ of $V$ is a \n$\\v
arphi$-Sidon set if\, \nfor all $h$-tuples $(a_1\,\\ldots\, a_h) \\in A^h$
and $ (a'_1\,\\ldots\, a'_h) \\in A^h$\, \nthe relation \n$\\varphi(a_
1\,\\ldots\, a_h) = \\varphi(a'_1\,\\ldots\, a'_h) \n$ implies $(a_1\,\\ld
ots\, a_h) = (a'_1\,\\ldots\, a'_h)$. \nThere exist infinite Sidon sets f
or the linear form $\\varphi$ if and only if the set of coefficients of $\
\varphi$ has distinct subset sums. \nIn a normed vector space with $\\var
phi$-Sidon sets\, \nevery infinite sequence of vectors is \nasymptotic to
a $\\varphi$-Sidon set of vectors.\nResults on $p$-adic perturbations of $
\\varphi$-Sidon sets of integers and bounds on the growth \nof $\\varphi$-
Sidon sets of integers are also obtained.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steve Miller (Williams College)
DTSTART;VALUE=DATE-TIME:20210218T200000Z
DTEND;VALUE=DATE-TIME:20210218T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/26
DESCRIPTION:Title: How low can we go? Understanding zeros of L
-functions near the central point\nby Steve Miller (Williams College)
as part of New York Number Theory Seminar\n\n\nAbstract\nSpacings between
zeros of $L$-functions occur throughout modern number theory\, \n such as
in Chebyshev's bias and the class number problem. Montgomery and Dyson \
n discovered in the 1970's that random matrix theory models these spacing
s. \n The initial models are insensitive to finitely many zeros\, and thu
s miss the behavior \n near the central point. This is the most arithmeti
cally interesting place\; for example\, \n the Birch and Swinnerton-Dyer
conjecture states that the rank of the Mordell-Weil group \n equals the o
rder of vanishing of the associated $L$-function there. To investigate the
zeros \n near the central point\, Katz and Sarnak developed a new statis
tic\, the $n$-level density\; \n one application is to bound the average
order of vanishing at the central point for a given \n family of $L$-func
tions by an integral of a weight against some test function $\\phi$. After
\n reviewing early results in the subject and describing how these stati
stics are computed\, \n we discuss as time permits recent progress and on
going work on several questions. \n We describe the Excised Orthogonal En
sembles and their success in explaining the \n observed repulsion of zero
s near the central point for families of $L$-functions\, \n and efforts t
o extend to other families. We discuss an alternative to the Katz-Sarnak \
n expansion for the $n$-level density which facilitate comparisons with r
andom matrix theory\,\n and applications to improving the bounds on high
vanishing at the central point. \n This work is joint with numerous summer
REU students.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guillermo Mantilla Soler (Universidad Konrad Lorenz\, Bogota\, Col
ombia)
DTSTART;VALUE=DATE-TIME:20210304T200000Z
DTEND;VALUE=DATE-TIME:20210304T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/27
DESCRIPTION:Title: Arithmetic equivalence and classification o
f number fields via the integral trace\nby Guillermo Mantilla Soler (
Universidad Konrad Lorenz\, Bogota\, Colombia) as part of New York Number
Theory Seminar\n\n\nAbstract\nTwo number fields are called arithmetically
equivalent if their Dedekind zeta functions coincide. Thanks to the work o
f R. Perlis\, we know that much of the arithmetic information of a number
field is encoded in its zeta function. By interpreting the Dedekind zeta f
unction as the Artin $L$-function attached to a certain Galois representa
tion of $G_{\\mathbb{Q}}$\, we see how all the information mentioned above
can be recovered in a very natural way. Moreover\, we will show how this
approach leads to new results. Going further\, we will see how from zeta
functions we can connect with trace forms and we will explore the classifi
cation power of integral trace forms.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thai Hoang Le (University of Mississippi)
DTSTART;VALUE=DATE-TIME:20210311T200000Z
DTEND;VALUE=DATE-TIME:20210311T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/28
DESCRIPTION:by Thai Hoang Le (University of Mississippi) as part of New Yo
rk Number Theory Seminar\n\nAbstract: TBA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pierre Bienvenue (Universite Claude Bernard Lyon)
DTSTART;VALUE=DATE-TIME:20210318T190000Z
DTEND;VALUE=DATE-TIME:20210318T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/29
DESCRIPTION:by Pierre Bienvenue (Universite Claude Bernard Lyon) as part o
f New York Number Theory Seminar\n\nAbstract: TBA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brandon Hanson (University of Georgia)
DTSTART;VALUE=DATE-TIME:20210325T190000Z
DTEND;VALUE=DATE-TIME:20210325T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/30
DESCRIPTION:Title: Sum-product and convexity\nby Brandon H
anson (University of Georgia) as part of New York Number Theory Seminar\n\
n\nAbstract\nA recurring theme in number theory is that addition and multi
plication do not mix well. \n A combinatorial take on this theme is the E
rdos-Szemeredi sum-product problem\, \n which says that a finite set of n
umbers (in an appropriate field) must have either a large \n sumset or a l
arge product set. Depending on the field one is working in\, ther
e are \n different tools which are useful for attacking this problem.
Over the real numbers\, \n convexity is one such tool. In this talk\, I
will discuss the sum-product problem and its\n variants\, and progress t
hat has been made on it. I will then discuss some elementary \n methods o
f using convexity to obtain some new results. This will all be based on r
ecent \n and ongoing work with P. Bradshaw\, O. Roche-Newton\, and M. Rudn
ev.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giorgis Petridis (University of Georgia)
DTSTART;VALUE=DATE-TIME:20210225T200000Z
DTEND;VALUE=DATE-TIME:20210225T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/31
DESCRIPTION:Title: Almost eventowns\nby Giorgis Petridis (
University of Georgia) as part of New York Number Theory Seminar\n\n\nAbst
ract\nLet $n$ be an even positive integer. An eventown is a collection of
subsets of $\\{1\,\\ldots\,n\\}$ \n with the property that every two not n
ecessarily distinct elements have even intersection. \n Berlekamp determin
ed the largest size of an even town in the 1960s\, answering \n a questio
n of Erd\\H{o}s. In line with other Erd\\H{o}s questions\, Ahmadi and Moha
mmadian \n made a conjecture on the size of the largest size of an almost
eventown: \n a family of subsets of $\\{1\, …\,n\\}$ with the property t
hat among any three elements \n there are two with even intersection. In t
his talk we will prove the conjecture and \n mention other related results
proved in joint work with Ali Mohammadi.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tim Trudgian (UNSW Canberra at the Australian Defense Force Academ
y)
DTSTART;VALUE=DATE-TIME:20210401T190000Z
DTEND;VALUE=DATE-TIME:20210401T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/32
DESCRIPTION:Title: Verifying the Riemann hypothesis to a new h
eight\nby Tim Trudgian (UNSW Canberra at the Australian Defense Force
Academy) as part of New York Number Theory Seminar\n\n\nAbstract\nSadly\,
I won't have time to prove the Riemann hypothesis in this talk. However\,
I do hope to outline recent work in a record partial-verification of RH. T
his is joint work with Dave Platt\, in Bristol\, UK.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (Lehman College (CUNY))
DTSTART;VALUE=DATE-TIME:20210408T190000Z
DTEND;VALUE=DATE-TIME:20210408T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/33
DESCRIPTION:Title: Inverse problems for Sidon sets\nby Mel
Nathanson (Lehman College (CUNY)) as part of New York Number Theory Semin
ar\n\n\nAbstract\nThe Riemann zeta function is an important function in nu
mber theory. It captures \n arithmetic properties of the integers. Riema
nn zeta values and multiple zeta values\, \n defined by Euler and Zagier\,
can be expressed in terms of iterated path integrals. \n Those iterated i
ntegrals a quite special. They have a very good meaning in terms \n of alg
ebraic geometry. More precisely\, the underlying algebraic variety is the
Deligne-Mumford comactification of the moduli space of curves of genus ze
ro. I will explain intuitively what that means. \n\n If we adjoin $\\sqrt
{2}$ or $i$ to the integers\, then the corresponding zeta functions are ca
lled Dedekind zeta functions. My main interest in this area is related to
the Dedekind \n zeta functions. I express them in terms of a higher dimen
sional iterated integrals\, \n which I call iterated integrals on membrane
s. Using this tool\, one can define multiple \n Dedekind zeta values as a
number theoretic analogue of multiple zeta values and \n relate them to a
lgebraic geometry and motives.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (Lehman College (CUNY))
DTSTART;VALUE=DATE-TIME:20210930T190000Z
DTEND;VALUE=DATE-TIME:20210930T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/34
DESCRIPTION:Title: Egyptian fractions and the Muirhead-Rado in
equality\nby Mel Nathanson (Lehman College (CUNY)) as part of New York
Number Theory Seminar\n\n\nAbstract\nFibonacci proved that a greedy algor
ithm constructs a representation of a positive rational number as the sum
of a finite number of Egyptian fractions. Sylvester used a greedy approx
imation algorithm to construct an increasing sequence of positive integers
$a_1\, a_2\, \\ldots$ such that $\\sum_{i=1}^n 1/a_i < 1$ and\, if $b_1\,
\\ldots\, b_n$ is any increasing sequence of positive integers such that
$\\sum_{i=1}^n 1/a_i \\leq \\sum_{i=1}^n 1/b_i < 1$\, then $a_i = b_i$ fo
r all $i = 1\,\\ldots\, n$. This result (conjectured by Kellogg and prove
d\, or believed to have been proved\, by several mathematicians) extends t
o Egyptian fraction approximations of other positive rational numbers. Th
e proof uses an application of the Muirhead inequality first observed by S
oundararajan.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (Lehman College (CUNY))
DTSTART;VALUE=DATE-TIME:20211007T190000Z
DTEND;VALUE=DATE-TIME:20211007T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/35
DESCRIPTION:Title: Problems and results on Egyptian fractions<
/a>\nby Mel Nathanson (Lehman College (CUNY)) as part of New York Number T
heory Seminar\n\n\nAbstract\nSome problems related to the theorem that Syl
vester's sequence (defined recursively by $a_0=1$\, $a_{n+1} = 1 +\\prod_
{i=1}^n a_i $) gives the best underapproximation to 1.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noah Lebowitz-Lockard (Philadelphia)
DTSTART;VALUE=DATE-TIME:20211014T190000Z
DTEND;VALUE=DATE-TIME:20211014T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/36
DESCRIPTION:Title: Binary Egyptian fractions\nby Noah Lebo
witz-Lockard (Philadelphia) as part of New York Number Theory Seminar\n\n\
nAbstract\nDefine a ``unit fraction" as a fraction with numerator $1$. We
say that an ``Egyptian fraction representation" of a number is a sum of di
stinct unit fractions. In this talk\, we discuss the history of these repr
esentations\, starting with their origins on an ancient Egyptian papyrus.
In particular\, we look at several recent results related to binary Egypti
an fractions\, which are sums of two unit fractions. Most of these results
relate to how often a given rational number has a binary Egyptian fractio
n representation.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Russell Jay Hendel (Towson University)
DTSTART;VALUE=DATE-TIME:20211021T190000Z
DTEND;VALUE=DATE-TIME:20211021T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/37
DESCRIPTION:Title: Limiting behavior of resistances in triangu
lar graphs\nby Russell Jay Hendel (Towson University) as part of New Y
ork Number Theory Seminar\n\n\nAbstract\nCertain electric circuit can be p
erceived as undirected graphs whose edges are 1-ohm resistances. \nOhm's
law allows calculation of equivalent single resistances \nbetween t
wo arbitrary points on the electric circuit. \nFor graphs embeddable in th
e plane\, there are four functions that allow the implementation of O
hm's law and \ncalculation of equivalent resistances. \nConsequently
\, no knowledge of electrical engineering is needed for this talk. \nIt
is a talk about interesting properties of graphs \nwhose edges have specif
ic resistances and \nwhich allow reduction to other graphs. \nInterest
ing results are possible when the underlying graph \nbelongs to certain
families. For example \nthe resistance between two corners \n(degree-tw
o vertices) of a graph on $n$\nedges consisting of $n-2$ triangles arra
nged in a line is \n$\\frac{n-1}{5}+ \\frac{4}{5} \\frac{F_{n-1}}{L_{n-1}}
$\nwith $F$ and $L$ representing the Fibonacci and Lucas numbers respect
ively\n\nThis presentation explores \ntriangular graphs of $n$ rows of equ
ilateral triangles. \nThese triangular graphs were mentioned in passing \n
in one paper with a conjecture on the equivalent resistance between \ntwo
corners. In this presentation we present new computation methods\, \nallow
ing reviewing more data. It turns out that the \nlimiting behavior of thes
e $n$-row triangular grids \n(as $n$ goes to infinity) has unexpected simp
ly described behavior: \nThe sides of individual triangles are conjecture
d to \nasymptotically equal products of basically \nfractional linear tran
sformations and $e^{-1}.$ \nWe also introduce new proof methods based on a
simple \n$verification$ method.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20211028T190000Z
DTEND;VALUE=DATE-TIME:20211028T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/38
DESCRIPTION:Title: Some results in elementary number theory\nby Mel Nathanson (CUNY) as part of New York Number Theory Seminar\n\n\n
Abstract\nVariations on Euler's totient function and associated arithmetic
identities.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ajmain Yamin (CUNY Graduate Center)
DTSTART;VALUE=DATE-TIME:20211104T190000Z
DTEND;VALUE=DATE-TIME:20211104T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/39
DESCRIPTION:Title: Complete regular dessins\nby Ajmain Yam
in (CUNY Graduate Center) as part of New York Number Theory Seminar\n\n\nA
bstract\nA map is an embedding of a graph into a topological surface such
that the complement of the image is a union of topological disks. A regul
ar map is one that exhibits the maximal amount of symmetry\, that is\, the
automorphism group of the map acts transitively on flags. In 1985\, James
and Jones classified complete regular maps\, i.e. regular maps where the
underlying graph is complete. The first goal of my talk is to give a brief
overview of this story and in particular review Biggs' construction of co
mplete regular maps as Cayley maps associated to finite fields. \n\n Given
any map\, one obtains a dessin by taking the bipartification of the under
lying graph and embedding that into the surface. Dessins associated to com
plete regular maps will be called \\emph{complete regular dessins} in my t
alk. After reviewing the basic theory of dessins\, I will introduce the ma
in question of my talk: can one obtain an explicit model for the Riemann s
urface underlying a complete regular dessin as an algebraic curve over $\\
mathbb{\\overline{Q}}$? What about the its Belyi function as a rational ma
p down to $\\mathbb{P}^1(\\mathbb{C})$? In this talk I will explain how to
obtain such an affine model for the complete regular dessin $K_5$ embedde
d in the torus. In the process\, we will be led to consider airithmetic i
n the Gaussian integers\, uniformization of elliptic curves\, Galois theor
y of function fields and Weierstrass $\\wp$ functions.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laszlo Toth (University of Pecs\, Hungary)
DTSTART;VALUE=DATE-TIME:20211111T200000Z
DTEND;VALUE=DATE-TIME:20211111T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/40
DESCRIPTION:Title: Menon's identity: proofs\, generalizations
and analogs\nby Laszlo Toth (University of Pecs\, Hungary) as part of
New York Number Theory Seminar\n\n\nAbstract\nMenon's identity states tha
t for every positive integer $n$ one has \n$\\sum (a-1\,n) = \\varphi(n) \
\tau(n)$\, where $a$ runs through a reduced residue system (mod $n$)\, \n$
(a-1\,n)$ stands for the greatest common divisor of $a-1$ and $n$\,\n$\\va
rphi(n)$ is Euler's totient function and $\\tau(n)$ is the number of divis
ors of $n$. It is named after Puliyakot Kesava Menon\, \nwho proved it in
1965. Menon's identity has been the subject of many research papers\, also
in the last years.\n\nIn this talk I will present different methods to pr
ove this identity\, and will point out those that I could not identify in
the literature. \nThen I will survey the directions to obtain generalizati
ons and analogs. I will also present some of my own general identities.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Cohen (MIT)
DTSTART;VALUE=DATE-TIME:20211202T200000Z
DTEND;VALUE=DATE-TIME:20211202T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/41
DESCRIPTION:Title: An optimal inverse theorem for tensors over
large fields\nby Alex Cohen (MIT) as part of New York Number Theory S
eminar\n\n\nAbstract\nA degree $k$ tensor $T$ over a finite field $\\mathb
f{F}_q$ can be viewed as a multilinear function $\\mathbf{F}_q^n \\times
\\dots \\times \\mathbf{F}_q^n \\to \\mathbf{F}_q.$\n The analytic rank of
$T$ takes a value between $0$ and $n$\, and is small if the output distri
bution is far from uniform---in some sense\, it is a measure of how random
ly $T$ behaves. On the other hand\, the partition rank of $T$ is small if
$T$ can be decomposed into a few highly structured pieces. It is not hard
to show that the analytic rank is less than the partition rank---or in oth
er words\, if $T$ is highly structured\, then it does not \n behave rando
mly. In 2008 Green and Tao proved a qualitative inverse theorem stating th
at the partition rank is bounded by some (large) function of the analytic
rank. We prove an \n optimal inverse theorem: Analytic rank and partitio
n rank are equivalent up to linear factors (over large enough fields). Th
is theorem allows us to explain any lack of randomness in $T$ by the pres
ence of structure. Our techniques are very different from the usual method
s in this area. We rely on algebraic geometry rather than additive combi
natorics. This is joint work with Guy Moshkovitz.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yunping Jiang (Queens College (CUNY))
DTSTART;VALUE=DATE-TIME:20211209T200000Z
DTEND;VALUE=DATE-TIME:20211209T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/42
DESCRIPTION:Title: Ergodic theory motivated by Sarnak's conjec
ture in number theory\nby Yunping Jiang (Queens College (CUNY)) as par
t of New York Number Theory Seminar\n\n\nAbstract\nSarnak's conjecture bri
ngs together number theory\, ergodic theory\, and dynamical systems. \n Mo
tivated by this conjecture\, we started a study in ergodic theory about or
ders of oscillating \n sequences and minimally mean attractable (MMA) and
minimally mean-L-stable (MMLS) flows. \n The Mobius function in number the
ory gives an example of oscillating sequences of order $d$ \n for all $d>0
$. From the dynamical systems point of view\, we found another class of ex
amples \n of oscillating sequences of order $d$ for all $d>0$. All equico
ntinuous flows are MMA and MMLA. \n I will talk about two non-trivial exam
ples of MMA and MMLS flows that are not equicontinuous. \n One is a Denjoy
counterexample in circle homeomorphisms and the other is an infinitely \n
renormalizable one-dimensional map. I will show that all oscillation sequ
ences of order 1\n are linearly disjoint with (or meanly orthogonal to) M
MA and MMLA flows. Thus\, we confirm \n Sarnak's conjecture for a large cl
ass of zero topological entropy flows. For oscillating sequences \n of ord
er $d>1$\, I will show that they are linearly disjoint from all affine dis
tal flows on the \n $d$-torus. One of the consequences is that Sarnak's co
njecture holds for all zero topological \n entropy affine flows on the $d$
-torus and some nonlinear zero topological entropy flows \n on the $d$-tor
us. I will also review some current developments after our work on this to
pic \n about flows with the quasi-discrete spectrum and the Thue-Morse seq
uence\, which has zero \n topological entropy and small Gowers norms and t
hus is a higher-order oscillating sequence.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guy Moshkovitz (Baruch College (CUNY))
DTSTART;VALUE=DATE-TIME:20211216T200000Z
DTEND;VALUE=DATE-TIME:20211216T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/43
DESCRIPTION:Title: An optimal inverse theorem for tensors over
large fields II\nby Guy Moshkovitz (Baruch College (CUNY)) as part of
New York Number Theory Seminar\n\n\nAbstract\nWe will give more details a
bout our recent proof\, joint with Alex Cohen\, showing that the partition
rank and the analytic rank of tensors are equal up to a constant\, over f
inite fields of every characteristic and of mildly large size (independent
of the number of variables). Proving the equivalence between these two qu
antities is a central question in additive combinatorics\, the main questi
on in the "bias implies low rank" line of work\, and corresponds to the fi
rst non-trivial case of the Polynomial Gowers Inverse conjecture.\n\nThe t
alk will be a continuation of Alex Cohen's talk from December 2nd\, though
I will aim for it to be mostly self-contained.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20220203T200000Z
DTEND;VALUE=DATE-TIME:20220203T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/44
DESCRIPTION:Title: Best underapproximation by Egyptian fractio
ns\nby Mel Nathanson (CUNY) as part of New York Number Theory Seminar\
n\n\nAbstract\nAn increasing sequence $(x_i)_{i=1}^n$ of positive integers
is an $n$-term Egyptian \nunderapproximation of $\\theta \\in (0\,1]$ if
$\\sum_{i=1}^n \\frac{1}{x_i} < \\theta$.\nA greedy algorithm constructs
an $n$-term underapproximation of $\\theta$. For some but not all number
s $\\theta$\, the greedy algorithm gives a unique best $n$-term underappr
oximation for all $n \\geq 1$. An infinite set of rational numbers is con
structed for which the greedy underapproximations are best\, and numbers
for which the greedy algorithm is not best are also studied.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20220210T200000Z
DTEND;VALUE=DATE-TIME:20220210T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/45
DESCRIPTION:Title: A chapter on the theory of equations: Desca
rtes\, Budan-Fourier\, and Sturm\nby Mel Nathanson (CUNY) as part of N
ew York Number Theory Seminar\n\n\nAbstract\nA discussion of the theorems
of Descartes\, Budan-Fourier\, and Sturm on the number of positive solutio
ns a polynomials equation in an interval $(a\,b]$. This is in preparatio
n for a discussion of Tarski's extension of Sturm's theorem and the Tarski
-Seidenberg decidability result.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Josiah Sugarman (CUNY Graduate Center)
DTSTART;VALUE=DATE-TIME:20220217T200000Z
DTEND;VALUE=DATE-TIME:20220217T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/46
DESCRIPTION:Title: The spectrum of the quaquaversal operator i
s real\nby Josiah Sugarman (CUNY Graduate Center) as part of New York
Number Theory Seminar\n\n\nAbstract\nIn the mid 90s Conway and Radin intro
duced the Quaquaversal Tiling. It is a hierarchical tiling of three dimens
ional space that exhibits statistical rotational symmetry\, in the sense t
hat the distribution of tiles chosen uniformly at random from a large sphe
re has a nearly uniform distribution of orientations. Any hierarchical til
ing has an associated operator whose spectrum can be analyzed to study the
distribution of orientations in a large sample. Radin and Conway showed t
hat 1 has multiplicity 1 in the spectrum of this operator to show that the
operator exhibited statistical rotational symmetry. By numerically analyz
ing the spectrum of this operator Draco\, Sadun\, and Wieren found eigenva
lues very close to 1 and concluded that the rate with which the distributi
on approaches uniformity is fairly slow\, mentioning that a galactic scale
sample of a material with this crystal structure at the molecular level w
ould exhibit noticeable anisotropy. Bourgain and Gamburd proved\, on the o
ther hand\, that a certain class of operators including this one have a no
nzero gap between 1 and the second largest eigenvalue\, concluding that th
e distribution must approach uniformity at an exponential rate.\n\nIn this
talk I will introduce hierarchical tilings\, discuss results similar to t
hose above\, and prove that the spectrum of this operator is real. Answeri
ng a question of Draco\, Sadun\, and Wieren.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20220224T200000Z
DTEND;VALUE=DATE-TIME:20220224T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/47
DESCRIPTION:Title: The Budan-Fourier theorem and multiplicity
matrices of polynomials\nby Mel Nathanson (CUNY) as part of New York N
umber Theory Seminar\n\n\nAbstract\nThe Budan-Fourier theorem gives an upp
er bound for the number of zeros \n (with multiplicity) of a polynomial $f
(x)$ of degree $n$ in the interval $(a\,b]$ \n in terms of the number of
sign variations in the vector of derivatives \n$D_f(\\lambda) = \\left( f
(\\lambda)\, f'(\\lambda)\, f''(\\lambda)\,\\ldots\, f^{(n)}(\\lambda) \\r
ight)$ \n at $\\lambda=a$ and $\\lambda=b$. \n One proof of the Budan-Fou
rier theorem considers the multiplicity vector \n $M_f(\\lambda) = \\left(
\\mu_0(\\lambda)\, \\mu_1(\\lambda)\, \\ldots\, \\mu_n(\\lambda) \\right
)$\, \nwhere $\\mu_j(\\lambda)$ is the multiplicity of $\\lambda$ as a roo
t \n of the $j$th derivative $f^{(j)}(x)$. \n The inverse problem asks: Wh
at vectors are the multiplicity vectors of polynomials\, \n and\, given a
multiplicity vector\, what are the associated polynomials? \n The simulta
neous study of multiplicities of real numbers $\\lambda_1\,\\ldots\, \\lam
bda_m$ leads to \n multiplicity matrices and their associated polynomial
s.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20220303T200000Z
DTEND;VALUE=DATE-TIME:20220303T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/48
DESCRIPTION:Title: Multiplicity matrices for polynomials\n
by Mel Nathanson (CUNY) as part of New York Number Theory Seminar\n\n\nAbs
tract\nLet $f(x)$ be a polynomial of degree $n$ and let $f^{(j)}(x)$ be th
e $j$th derivative of $f(x)$.\n Let $\\Lambda = (\\lambda_1\,\\ldots\, \\l
ambda_m)$ be a strictly increasing sequence of real numbers. \n For $i \\
in \\{1\,\\ldots\, m\\}$ and $j \\in \\{0\,1\,\\ldots\, n\\}$\, \nlet $ \
\mu_{i\,j}$ be the multiplicity of $\\lambda_i$ as a root \n of the polyno
mial $f^{(j)}(x)$. For $i \\in \\{1\,\\ldots\, m\\}$ and $j \\in \\{0\,1\
,\\ldots\, n\\}$\, \nlet $ \\mu_{i\,j}$ be the \n multiplicity of $\\lam
bda_i$ as a root of the polynomial $f^{(j)}(x)$. \nThe multiplicity matrix
of $f$ \n with respect to $\\lambda_1\,\\ldots\, \\lambda_m$\nis the $m
\\times (n+1)$ matrix \n$\nM_f(\\Lambda) = \n\\begin{matrix} \\mu_{i\,j}
\n\\end{matrix}.\n$\n The problem is to describe the matrices are multip
licity matrices of polynomials.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noah Kravitz (Princeton University)
DTSTART;VALUE=DATE-TIME:20220310T200000Z
DTEND;VALUE=DATE-TIME:20220310T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/49
DESCRIPTION:Title: Multiplicity matrices and zeros of polynomi
als\nby Noah Kravitz (Princeton University) as part of New York Number
Theory Seminar\n\n\nAbstract\nEarlier this week\, Nathanson introduced th
e notion of the derivative matrix associated with a polynomial and a finit
e tuple of points. He established several properties of derivative matric
es and proposed a number of appealing open problems. I will discuss Natha
nson's setup\, the solutions to a few of his problems\, and partial progre
ss on natural follow-up questions.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:David and Gregory Chudnovsky (NYU Tandon School of Engineering)
DTSTART;VALUE=DATE-TIME:20220317T190000Z
DTEND;VALUE=DATE-TIME:20220317T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/50
DESCRIPTION:Title: How to break step\nby David and Gregory
Chudnovsky (NYU Tandon School of Engineering) as part of New York Number
Theory Seminar\n\nAbstract: TBA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Itay Londner (Weizmann Institute of Science\, Israel)
DTSTART;VALUE=DATE-TIME:20220324T190000Z
DTEND;VALUE=DATE-TIME:20220324T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/51
DESCRIPTION:Title: Tiling the integers with translates of one
tile: the Coven-Meyerowitz tiling conditions\nby Itay Londner (Weizman
n Institute of Science\, Israel) as part of New York Number Theory Seminar
\n\n\nAbstract\nIt is well known that if a finite set of integers A tiles
the integers by translations\, then the translation set must be periodic\
, so that the tiling is equivalent to a factorization $A+B=Z_M$ of a fini
te cyclic group. Coven and Meyerowitz (1998) proved that when the tiling p
eriod $M$ has at most two distinct prime factors\, each of the sets A and
B can be replaced by a highly ordered "standard" tiling complement. It is
not known whether this behavior persists for all tilings with no restricti
ons on the number of prime factors of $M$. In joint work with Izabella La
ba (UBC)\, we proved that this is true for all sets tiling the integers wi
th period $M=(pqr)^2$. In my talk I will discuss this problem and introduc
e some ideas from the proof.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Darij Grinberg (Drexel University)
DTSTART;VALUE=DATE-TIME:20220331T190000Z
DTEND;VALUE=DATE-TIME:20220331T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/52
DESCRIPTION:Title: From the Vandermonde determinant to general
ized factorials to greedoids and back\nby Darij Grinberg (Drexel Unive
rsity) as part of New York Number Theory Seminar\n\n\nAbstract\nA classica
l result in elementary number theory says that the\nproduct of the pairwis
e \n differences between any given $n + 1$ integers\nis divisible by the p
roduct of the pairwise \n differences between $0\, 1\,\n...\, n$. In the l
ate 90s\, Manjul Bhargava developed this much further\n into a theory of "
generalized factorials\," in particular giving a\nquasi-algorithm for find
ing \n the gcd of the products of the pairwise\ndifferences between any $
n + 1$ integers in $S$\, \n where $n$ is a given number\nand $S$ is a give
n set of integers.\nIn this talk\, I will explain \n why this is actually
a combinatorial\nquestion in disguise\, and how to answer it in full \n ge
nerality (joint\nwork with Fedor Petrov). The general setting is a finite
set $E$\nequipped \n with weights (every element of $E$ has a weight) and
distances\n(any two distinct elements \n of $E$ have a distance)\, where t
he distances\nsatisfy the ultrametric triangle inequality. \n The question
is then to\nfind a subset of $E$ of given size that has maximum perimeter
\n (i.e.\, sum\nof weights of elements plus their pairwise distances). It
turns out\nthat all such \n subsets form a "strong greedoid" -- a type of
set system\nparticularly adapted to optimization. \n Even better\, this g
reedoid is a\n"Gaussian elimination greedoid" -- which\, roughly speaking\
, \n means that\nthe problem reduces to linear algebra.\nIf time allows\,
I will briefly discuss \n another closely related\ngreedoid coming from a
rather similar problem in phylogenetics. \n (This\nis mostly due to Manson
\, Moulton\, Semple and Steel.)\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giorgis Petridis (University of Georgia)
DTSTART;VALUE=DATE-TIME:20220414T190000Z
DTEND;VALUE=DATE-TIME:20220414T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/54
DESCRIPTION:Title: On a question of Yufei Zhao on the interfac
e of combinatorial geometry\nby Giorgis Petridis (University of Georgi
a) as part of New York Number Theory Seminar\n\n\nAbstract\nLet $A$ be a f
inite set of integers and consider the lines determined by pairs of points
of $P = \\{(a\,a^2) : a \\in A\\}$. The sum set of $A$ is the set of slop
es of these lines and the product set of $A$ is the set of $y$-intercepts.
We know from the celebrated sum-product theorem of Erd\\H{o}s and Szemer\
\'edi that at least one of these sets is much larger than $|A|$. Geometric
ally\, this observation can be phrased as follows: infinity cannot both b
e close to the minimum. Motivated by this observation\, Yufei Zhao asked i
f this is a manifestation of a more general phenomenon. The goal of the ta
lk is to answer this in the affirmative. Joint work with O. Roche-Newton\
, M. Rudnev and A. Warren.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20220407T190000Z
DTEND;VALUE=DATE-TIME:20220407T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/55
DESCRIPTION:Title: Exponential automorphisms and a problem of
Mycielski\nby Mel Nathanson (CUNY) as part of New York Number Theory S
eminar\n\n\nAbstract\nAn exponential automorphism of $\\mathbf{C}$ is a fu
nction $\\alpha: \\mathbf{C} \\rightarrow \\mathbf{C}$ such that \n$\\alp
ha(z + w) = \\alpha(z) + \\alpha(w)$\nand\n$\\alpha\\left( e^z \\right) =
e^{\\alpha(z)}$\nfor all $z\, w \\in \\C$. \nMycielski asked if $\\alpha(\
\log 2) = \\log 2$ and if $\\alpha(2^{1/k}) = 2^{1/k}$ for $k = 2\, 3\, 4$
.\nThis paper solves these problems.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20220428T190000Z
DTEND;VALUE=DATE-TIME:20220428T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/56
DESCRIPTION:Title: Multiplicity interpolation and the theorems
of Descartes and Budan-Fourier\nby Mel Nathanson (CUNY) as part of Ne
w York Number Theory Seminar\n\nAbstract: TBA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20220505T190000Z
DTEND;VALUE=DATE-TIME:20220505T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/57
DESCRIPTION:Title: Polynomials and the Budan-Fourier theorem\nby Mel Nathanson (CUNY) as part of New York Number Theory Seminar\n\nA
bstract: TBA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20220512T190000Z
DTEND;VALUE=DATE-TIME:20220512T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/58
DESCRIPTION:Title: van der Waerden's proof of Sturm's theorem<
/a>\nby Mel Nathanson (CUNY) as part of New York Number Theory Seminar\n\n
\nAbstract\nContinuation of series of talks on classical results for count
ing the number of real roots of polynomials with real coefficients.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steven J. Miller (Williams College)
DTSTART;VALUE=DATE-TIME:20220616T190000Z
DTEND;VALUE=DATE-TIME:20220616T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/59
DESCRIPTION:Title: Benford's Law: Why the IRS might care about
the 3x+1 problem and zeta(s)\nby Steven J. Miller (Williams College)
as part of New York Number Theory Seminar\n\n\nAbstract\nMany systems exhi
bit a digit bias. For example\, the first digit base 10 of the \n Fibonacc
i numbers or of $2^n$ equals 1 about 30\\% of the time\; the IRS uses this
digit bias to detect fraudulent corporate tax returns. This phenomenon\,
\n known as Benford's Law\, was first noticed by observing which pages of
log tables \n were most worn from age -- it's a good thing there were no c
alculators 100 years ago! \n We'll discuss the general theory and applica
tion\, talk about some fun examples \n (ranging from the $3x+1$ problem to
the Riemann zeta function to fragmentation \n problems\, as time permits)
\, and see how the irrationality type of numbers often \n enter into the a
nalysis (through error terms in equidistribution theorems).\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20220623T190000Z
DTEND;VALUE=DATE-TIME:20220623T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/60
DESCRIPTION:Title: Arithmetic functions and fixed points of po
wers of permutations\nby Mel Nathanson (CUNY) as part of New York Numb
er Theory Seminar\n\n\nAbstract\nLet $\\sigma$ be a permutation of a finit
e or infinite set $X$\, \nand let $F_X\\left( \\sigma^k\\right)$ count th
e number of fixed points of \nthe $k$th power of $\\sigma$.\nThis paper de
scribes how the sequence $\\left(F_X\\left( \\sigma^k\\right) \\right)_{k=
1}^{\\infty}$ \ndetermines the conjugacy class of the permutation $\\sigma
$. \nWe also describe the arithmetic functions that are fixed point sequ
ences of permutations.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20220630T190000Z
DTEND;VALUE=DATE-TIME:20220630T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/61
DESCRIPTION:Title: Continuity of the roots of a polynomial
\nby Mel Nathanson (CUNY) as part of New York Number Theory Seminar\n\n\nA
bstract\nLet $K$ be an algebraically closed field with an absolute value.
We give an elementary \n (high school algebra) proof of the classical res
ult that the roots of a polynomial \n with coefficients in $K$ are contin
uous functions of the coefficients of the polynomial. \n \n Joint work wi
th David Ross.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:David A. Ross (University of Hawaii)
DTSTART;VALUE=DATE-TIME:20220707T190000Z
DTEND;VALUE=DATE-TIME:20220707T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/62
DESCRIPTION:Title: Yet another proof that the roots of a polyn
omial depend continuously on the coefficients\nby David A. Ross (Unive
rsity of Hawaii) as part of New York Number Theory Seminar\n\n\nAbstract\n
The roots of a complex polynomial depend continuously on the coefficients\
; that is\, an infinitesimal perturbation of the coefficients results in a
n infinitesimal perturbation of the roots. I'll give a short\, straightf
orward proof of this using infinitesimals.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20220714T190000Z
DTEND;VALUE=DATE-TIME:20220714T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/63
DESCRIPTION:Title: A nonstandard proof of continuity of affine
varieties\nby Mel Nathanson (CUNY) as part of New York Number Theory
Seminar\n\n\nAbstract\nExtending the classical result \nthat the roots of
a polynomial with coefficients in $\\mathbf{C}$ are continuous functions \
nof the coefficients of the polynomial\, nonstandard analysis is used to p
rove that \nif $\\mathcal{F} = \\{f_{\\lambda} :\\lambda \\in \\Lambda\\
}$ \nis a set of polynomials in $\\C[t_1\,\\ldots\, t_n]$ and if \n $^*\\m
athcal{G} = \\{g_{\\lambda} :\\lambda \\in \\Lambda\\}$ \n is a set of po
lynomials in $^*\\mathbf{C}_0[t_1\,\\ldots\, t_n]$ \n such that $g_{\\lamb
da}$ is an infinitesimal deformation of $f_{\\lambda}$ \n for all $\\lambd
a \\in \\Lambda$\, \n then the nonstandard affine variety $^*V_0(\\mathca
l{G})$ \n is an infinitesimal deformation of the affine variety $V(\\mathc
al{F})$.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin O'Bryant (College of Staten Island (CUNY))
DTSTART;VALUE=DATE-TIME:20220721T190000Z
DTEND;VALUE=DATE-TIME:20220721T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/64
DESCRIPTION:Title: On the size of finite Sidon sets\nby Ke
vin O'Bryant (College of Staten Island (CUNY)) as part of New York Number
Theory Seminar\n\n\nAbstract\nIn 2021\, Balogh-Furedi-Roy proved that any
Sidon set with $k$ elements has diameter at least $k^2-1.996 k^{3/2}$\, pr
ovided that $k$ is sufficiently large. We give a method logically simple
r than the BFR one\, though trading on the same phenomena\, but substantiv
ely more involved computationally. The diameter of a $k$ element Sidon set
is at least $k^2 - 1.99405 k^{3/2}$.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ali Khan (Johns Hopkins University)
DTSTART;VALUE=DATE-TIME:20220728T190000Z
DTEND;VALUE=DATE-TIME:20220728T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/65
DESCRIPTION:Title: Optimal chaotic dynamics\, the checkmap and
the 2-sector RSS model\nby Ali Khan (Johns Hopkins University) as par
t of New York Number Theory Seminar\n\n\nAbstract\nThis talk reports on jo
int work with Deng\, Fujio and Rajan on optimal chaotic dynamics in mathem
atical economics revolving around the check-map and a model due to Robinso
n-Srinivasan-Solow – the RSS model. I hope to emphasize number-theoretic
considerations\, and touch on earlier work of Nathanson (1976 PAMS)\, and
more conjecturally with two papers of Lagarias and co-authors (J. London
Math. Soc. 1993\; Ill. J. Math. 1994) on the asymmetric tent map. Geomet
ry as an engine of analysis will also be emphasized.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20220804T190000Z
DTEND;VALUE=DATE-TIME:20220804T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/66
DESCRIPTION:Title: Patterns in the iteration of an arithmetic
function\nby Mel Nathanson (CUNY) as part of New York Number Theory Se
minar\n\n\nAbstract\nLet $\\Omega$ be a set of positive integers and let $
S:\\Omega \\rightarrow \\Omega$\n be an arithmetic function. Let $V = (v_
i)_{i=1}^n$ be a finite sequence of positive integers. \nAn integer $m \\
in \\Omega$ has increasing-decreasing pattern $V$ with respect to $S$ if\,
\nfor odd integers $i \\in \\{1\,\\ldots\, n\\}$\, \n\\[\nS^{v_1+ \\cdo
ts + v_{i-1}}(m) < S^{v_1+ \\cdots + v_{i-1}+1}(m) < \\cdots < S^{v_1+ \\c
dots + v_{i-1}+v_{i}}(m)\n\\]\nand\, for even integers $i \\in \\{2\,\\ld
ots\, n\\}$\, \n\\[\nS^{v_1+ \\cdots + v_{i-1}}(m) > S^{v_1+ \\cdots +v_{
i-1}+1}(m) > \\cdots > S^{v_1+ \\cdots +v_{i-1}+v_i}(m).\n\\]\nThe arithm
etic function $S$ is wildly increasing-decreasing if\, \nfor every finite
sequence $V$ of positive integers\, there exists an integer $m \\in \\Om
ega$ \nsuch that $m$ has increasing-decreasing pattern $V$ with respect to
$S$. \nThis paper gives a new proof that the Collatz function \nis wildl
y increasing-decreasing.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20220929T190000Z
DTEND;VALUE=DATE-TIME:20220929T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/67
DESCRIPTION:Title: Poincare's Positivstellensatz\nby Mel N
athanson (CUNY) as part of New York Number Theory Seminar\n\n\nAbstract\nP
oincare's proof of Poincare's theorem (H. Poincare\, Sur les equations a
lgebriques\, Comptes Rendus 97 (1883)\, 1418--1419).\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin O'Bryant (CUNY\, College of Staten Island and The Graduate C
enter)
DTSTART;VALUE=DATE-TIME:20221027T190000Z
DTEND;VALUE=DATE-TIME:20221027T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/68
DESCRIPTION:Title: Finite Sidon sets\nby Kevin O'Bryant (C
UNY\, College of Staten Island and The Graduate Center) as part of New Yor
k Number Theory Seminar\n\n\nAbstract\nA finite Sidon set is a set $A = \\
{ a_1 < a_2 < ... < a_k \\}$ with all the sums $a_i+a_j$ with $i \\leq j$
different. We will review the history of Sidon sets before turning \n our
attention to recent progress bounding the diameter of $A$ in terms of the
size\n of $A$. This talk is suitable for tourists and newcomers to additiv
e number theory.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20221117T200000Z
DTEND;VALUE=DATE-TIME:20221117T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/69
DESCRIPTION:Title: Von Neumann's decomposition of intervals in
to countably infinitely many pairwise disjoint and congruent subsets\n
by Mel Nathanson (CUNY) as part of New York Number Theory Seminar\n\n\nAbs
tract\nAn exposition of von Neumann's paper\, ``Die Zerlegung eines Inter
valles in abzahlbar viele kongruente Teilmengen'' (Fund. Math. 11 (1928)\,
230--238)\, of which Freeman Dyson wrote\, ``In another corner of [Johnny
von Neumann's] garden\, there is a little flower all by itself\, a short
paper ... [that] solves a problem raised by the Polish mathematician Hugo
Steinhaus....''\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Iosevich (University of Rochester)
DTSTART;VALUE=DATE-TIME:20230209T200000Z
DTEND;VALUE=DATE-TIME:20230209T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/71
DESCRIPTION:Title: Finite point configurations and complexity<
/a>\nby Alex Iosevich (University of Rochester) as part of New York Number
Theory Seminar\n\n\nAbstract\nWe are going to discuss connections between
the notion of the Vapnik-Chervonenkis dimension and some classical Erdos
-type problems.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhi-Wei Sun (Nanjing University\, P. R. China)
DTSTART;VALUE=DATE-TIME:20230216T200000Z
DTEND;VALUE=DATE-TIME:20230216T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/72
DESCRIPTION:Title: New results on power residues modulo primes
\nby Zhi-Wei Sun (Nanjing University\, P. R. China) as part of New Yor
k Number Theory Seminar\n\n\nAbstract\nIn this talk we introduce some new
results on power residues modulo primes.\n\nLet $p$ be an odd prime\, and
let $a$ be an integer not divisible by $p$.\nWhen $m$ is a positive intege
r with $p\\equiv 1\\pmod{2m}$ and $2$ is an $m$th power residue modulo $p$
\,\nthe speaker determines the value of the product $\\prod_{k\\in R_m(p)}
(1+\\tan\\pi\\frac{ak}p)$\, where\n$R_m(p)=\\{03$ be a
prime.\nLet $b\\in\\mathbb Z$ and $\\varepsilon\\in\\{\\pm1\\}$.\nJoint w
ith Q.-.H. Hou and H. Pan\, we prove that\n$\\left|\\left\\{N_p(a\,b):\\ 1
\\{ax^2+b\\}_p$\, and\n$\\{m\\}_p$
with $m\\in\\mathbb Z$ is the least nonnegative residue of $m$ modulo $p$.
\n\nWe will also mention some open conjectures.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hung Viet Chu (University of Illinois)
DTSTART;VALUE=DATE-TIME:20230223T200000Z
DTEND;VALUE=DATE-TIME:20230223T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/73
DESCRIPTION:Title: Underapproximation by Egyptian fractions an
d the weak greedy algorithm\nby Hung Viet Chu (University of Illinois)
as part of New York Number Theory Seminar\n\n\nAbstract\nNathanson recent
ly studied the greedy underapproximation algorithm which\, given $\\theta\
\in (0\,1]$\, produces a sequence of positive integers $(a_n)_{n=1}^\\inft
y$ such that $\\sum_{n=1}^\\infty 1/a_n = \\theta$. The algorithm is ``gre
edy" in the sense that at each step\, $a_n$ is chosen to be the smallest p
ositive integer such that \n$$\\frac{1}{a_n} \\ <\\ \\theta-\\sum_{i=1}^{n
-1}\\frac{1}{a_i}.$$\n\n\nWe introduce the weak greedy underapproximation
algorithm (WGUA)\, which follows the ``greedy choice up to a constant." In
particular\, for each $\\theta$\, the WGUA produces two sequences of posi
tive integers $(a_n)$ and $(b_n)$ such that \n\na) $\\sum_{n=1}^\\infty 1/
b_n = \\theta$\;\n\nb) $1/a_{n+1} < \\theta - \\sum_{i=1}^{n}1/b_i < 1/(a_
{n+1}-1)$ for all $n\\geqslant 1$\;\n\nc) there exists $t\\geqslant 1$ suc
h that $b_n/a_n \\leqslant t$ infinitely often.\n\nA sequence of positive
integers $(b_n)_{n=1}^\\infty$ is called a weak greedy underapproximation
of $\\theta$ if $\\sum_{n=1}^{\\infty}1/b_n = \\theta$.\nWe investigate wh
en a given weak greedy underapproximation $(b_n)$ can be produced by the W
GUA. Furthermore\, we show that for any increasing $(a_n)$ with $a_1\\geqs
lant 2$ and $a_n\\rightarrow\\infty$\, there exist $\\theta$ and $(b_n)$ s
uch that a) and b) are satisfied\; whether c) is also satisfied depends on
the sequence $(a_n)$. Finally\, we address the uniqueness of $\\theta$ an
d $(b_n)$ and apply our framework to specific sequences.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20230302T200000Z
DTEND;VALUE=DATE-TIME:20230302T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/74
DESCRIPTION:Title: Sinkhorn limits for generalized doubly sto
chastic matrices and tensors\nby Mel Nathanson (CUNY) as part of New Y
ork Number Theory Seminar\n\n\nAbstract\nSinkhorn's theorem asserts that i
f $A$ is a square matrix with positive coordinates\, then there \nexist (e
ssentially unique) positive diagonal matrices $X$ and $Y$ such that $XAY$
is doubly stochastic. \nMenon applied Brouwer's fixed point theorem to pr
ove this result. This talk will describe Menon's method and an extension
to higher dimensional tensors.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Federico Glaudo (Institute for Advanced Study)
DTSTART;VALUE=DATE-TIME:20230330T190000Z
DTEND;VALUE=DATE-TIME:20230330T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/75
DESCRIPTION:Title: Can you determine a set from its subset sum
s?\nby Federico Glaudo (Institute for Advanced Study) as part of New Y
ork Number Theory Seminar\n\n\nAbstract\nLet $A$ be a multiset with elemen
ts in an abelian group. \nLet $FS(A)$ be its subset sums \n multiset\, i.
e.\, the multiset containing the $2^{|A|}$ sums of all subsets of $A$. \n
Given $FS(A)$\, can you determine $A$? \n\n\n If the abelian group is $
\\Z$\, one can see that the two multisets $A=\\{-2\, 1\, 1\\}$ \n and $A'=
\\{-1\,-1\,2\\}$ satisfy $FS(A)=FS(A')$\; notice that one is obtained from
the other \n by changing signs to the elements. We will see that this is
the only obstruction and so\, \n up to the sign of the elements\, $FS(A)$
determines $A$ in $\\Z$.\n\nIn a general abelian group the situation is mu
ch more involved and we will see that the \n answer depends intimately o
n the orders of the torsion elements of the group.\n \n The core
of the proof relies on a delicate study of the structure of cyclotomic un
its \n and on an inversion formula for a novel discrete Radon transform
on finite abelian groups.\n \n This is a joint work with Andrea Cip
rietti.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Russell Jay Hendel (Towson University)
DTSTART;VALUE=DATE-TIME:20230309T200000Z
DTEND;VALUE=DATE-TIME:20230309T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/76
DESCRIPTION:Title: Recursions\, closed forms\, and characteris
tic polynomials of the circuit array\nby Russell Jay Hendel (Towson Un
iversity) as part of New York Number Theory Seminar\n\n\nAbstract\nOne mod
ern graph metric represents an electrical circuit with a graph whose edge
s are replaced with resistors and the so-called resistance distance betwee
n the nodes is determined by calculating the electrical resistance in the
circuit. Electrical circuit theory provides functions that allow ``reducti
on'' of one circuit to another circuit where the resistance distance betwe
en certain vertices is preserved. Recently there has been study of a graph
\, representable in the Cartesian plan as an n-grid\, n rows of upright eq
uilateral triangles\, all of whose edges are labeled one. It is possible t
o reduce the n-grid to an (n-1)-grid with resistance preserving operations
. The collections of successive reductions has many interesting properties
. In this talk we continue to study a ``slice'' of this collection of grid
s represented by the Circuit Array\, an infinite array of rational functio
ns. We show that certain closed forms\, recursions\, and characteristic po
lynomials (annihilators) emerge. One surprising result is that the annihil
ators of the numerators and denominators of the underlying rational functi
ons exclusively have roots which are integral powers of 9.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashvin Rajan
DTSTART;VALUE=DATE-TIME:20230420T190000Z
DTEND;VALUE=DATE-TIME:20230420T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/77
DESCRIPTION:Title: A diophantine problem on products of three
consecutive integers\nby Ashvin Rajan as part of New York Number Theor
y Seminar\n\n\nAbstract\nWe prove that (3\,4\,5) is the only triple of con
secutive positive integers whose product when doubled also factors as a pr
oduct of three consecutive integers. Joint work with Francois Ramaroson.\
n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sayak Sengupta (SUNY-Binghamton)
DTSTART;VALUE=DATE-TIME:20230504T190000Z
DTEND;VALUE=DATE-TIME:20230504T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/78
DESCRIPTION:Title: Locally nilpotent polynomials over Z\nb
y Sayak Sengupta (SUNY-Binghamton) as part of New York Number Theory Semin
ar\n\n\nAbstract\nLet $K$ be a number field and $\\mathcal{O}_K$ be the ri
ng of integers of $K$. For a polynomial $u(x)$ in $\\mathcal{O}_K[x]$ and
$r\\in\\mathcal{O}_K$\, we can construct a dynamical sequence $u(r)\,u^{(2
)}(r)\,\\ldots$. Let $P(u^{(n)}(r)):=\\{\\mathfrak{p}\\in \\text{MSpec}(\\
mathcal{O}_K)~|~u^{(n)}(r)\\in \\mathfrak{p}\,\\text{for some }n\\in\\math
bb{N} \\}$. For which polynomials $u(x)$ and $r\\in \\mathcal{O}_K$ do we
expect to have $P(u^{(n)}(r))=\\text{MSpec}(\\mathcal{O}_K)$? If we hit 0
somewhere in the above sequence\, then we obviously have the equality. If
we do not hit zero for any iteration then the question becomes very intere
sting. In this talk\, we will define such polynomials for a general number
field $K$ and then we will look at some results in the particular case of
$K=\\mathbb{Q}.$ This talk is based on a preprint of my ongoing work\, wh
ich is available in arXiv under the same name as the title.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Shallit (University of Waterloo)
DTSTART;VALUE=DATE-TIME:20230427T190000Z
DTEND;VALUE=DATE-TIME:20230427T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/79
DESCRIPTION:Title: Doing additive number theory with logic and
automata\nby Jeffrey Shallit (University of Waterloo) as part of New
York Number Theory Seminar\n\n\nAbstract\nThe classical tools of the addit
ive number theorist include analytic\nand combinatorial methods\, such as
the circle method and the sieve method.\nIn this talk I will present anoth
er method\, based on logic and automata\ntheory\, that can sometimes be us
ed to prove results in additive number\ntheory in relatively simple ways.
No experience with finite automata is \nassumed.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20240201T200000Z
DTEND;VALUE=DATE-TIME:20240201T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/80
DESCRIPTION:Title: Finitely many implies infinitely many (for
polynomials in infinitely many variables)\nby Mel Nathanson (CUNY) as
part of New York Number Theory Seminar\n\n\nAbstract\nMany mathematical s
tatements have the following form: Let $X$ be an infinite set of equation
s. If every finite subset of the equations has a common solution\, then
the infinite set of equations has a common solution. A result of this t
ype will be described for certain infinite sets of polynomial equations i
n infinitely many variables. \n\n This is joint work with David Ross.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Ross (University of Hawaii)
DTSTART;VALUE=DATE-TIME:20240208T200000Z
DTEND;VALUE=DATE-TIME:20240208T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/81
DESCRIPTION:Title: Finitely many implies infinitely many\, par
t 3: the nonstandard version\nby David Ross (University of Hawaii) as
part of New York Number Theory Seminar\n\n\nAbstract\nIn a pair of recent
seminars\, Mel Nathanson has discussed proofs\, using the Tychonoff Theore
m\, for existence of solutions to infinite sets of equations in infinitely
many variables. In at least one case the proof was an adaptation of an ar
gument using nonstandard analysis. In this talk I'll try to explain such n
onstandard arguments\, hopefully making them intelligible to mathematician
s who haven't seen nonstandard methods before.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Florian Luca (Wits and Oxford)
DTSTART;VALUE=DATE-TIME:20240215T200000Z
DTEND;VALUE=DATE-TIME:20240215T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/82
DESCRIPTION:Title: Positive integers $k$ such that $3^k+1\\equ
iv 0\\pmod {3k+1}$\nby Florian Luca (Wits and Oxford) as part of New Y
ork Number Theory Seminar\n\n\nAbstract\nIn my talk we will look at positi
ve integers $k$ such that $3^k+1\\equiv 0\\pmod {3k+1}$. We show that ther
e are infinitely many such. They are all odd and composite and they have a
counting function that is much smaller than the primes. This is work in p
rogress.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sayak Sengupta (Binghamton University (SUNY))
DTSTART;VALUE=DATE-TIME:20240222T200000Z
DTEND;VALUE=DATE-TIME:20240222T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/83
DESCRIPTION:Title: Nilpotent and infinitely nilpotent integer
sequences\nby Sayak Sengupta (Binghamton University (SUNY)) as part of
New York Number Theory Seminar\n\n\nAbstract\nWe say that an integer sequ
ence $\\{r_n\\}_{n\\ge 0}$ has a generating polynomial $u(x)$ over $\\math
bb{Z}$ if for every positive integer $n$ one has $u^{(n)}(r_0)=r_n$. In ad
dition\, if such a sequence satisfies the condition that $r_n=0$ for some
positive integer $n$ (respectively\, $r_n=0$ for infinitely many positive
integers $n$)\, then we say that $\\{r_n\\}_{n\\ge 0}$ is a nilpotent sequ
ence (respectively\, $\\{r_n\\}_{n\\ge 0}$ is an infinitely nilpotent sequ
ence). In this talk we will provide (and discuss) some important character
istics of nilpotent and infinitely nilpotent sequences.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Senia Sheydvasse (Bates College)
DTSTART;VALUE=DATE-TIME:20240229T200000Z
DTEND;VALUE=DATE-TIME:20240229T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/84
DESCRIPTION:Title: Hidden structures in families of Ulam seque
nces\nby Senia Sheydvasse (Bates College) as part of New York Number T
heory Seminar\n\n\nAbstract\nStanislaw Ulam defined the original Ulam sequ
ence as follows: Start with 1\,2\, and then each subsequent term is the ne
xt smallest integer that is the sum of two distinct prior terms in exactly
one way. (The next few terms are 1\,2\,3\,4\,6\,8\,...) There is now a ve
ritable zoo of "Ulam-like" sequences and sets\, most of which share the ma
in trait of the original: there is clear numerical evidence that there is
an underlying structure\, but for the most part we can prove almost nothin
g. (As a simple example: computation of trillions of terms of the Ulam seq
uence strongly suggests that it grows linearly. The best known bound is th
at it can't grow faster than exponentially fast.) One of the few partial r
esults that we can prove concerns what has been termed the Rigidity Conjec
ture. The original proofs surrounding this were model-theoretic in nature-
--what we shall show is that there is a completely constructive proof usin
g a new variation of Ulam sequences\, and the hints toward a broader solut
ion that this offers.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Sellers (University of Minnesota - Duluth)
DTSTART;VALUE=DATE-TIME:20240307T200000Z
DTEND;VALUE=DATE-TIME:20240307T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/85
DESCRIPTION:Title: Surprising connections between integer part
itions statistics: The crank\, minimal excludant\, and partition fixed poi
nts\nby James Sellers (University of Minnesota - Duluth) as part of Ne
w York Number Theory Seminar\n\n\nAbstract\nA {\\it partition} of an integ
er $n$ is a finite sequence of positive integers $p_1\\geq p_2\\geq \\dot
s \\geq p_k$ such that $n=p_1+p_2+\\dots + p_k.$ We let $p(n)$ denote the
number of partitions of $n$. For example\, $p(4) = 5$ because there are
five partitions of the integer $n=4$: \n\n$$4\, \\ \\ 3+1\, \\ \\ 2+2\, \
\ \\ 2+1+1\, \\ \\ 1+1+1+1$$\n\nIn 1919\, just one year before his death\,
Ramanujan discovered and proved some unexpected\, and truly amazing\, div
isibility properties for the function $p(n).$ Since then\, several mathem
aticians have studied $p(n)$ from different perspectives\, trying to bette
r understand these divisibility properties\, especially from a combinatori
al perspective. In the process\, numerous ``statistics'' have been define
d on partitions\, including the rank and crank of a partition. In this ta
lk\, I will discuss this history in more detail\, and then I will transiti
on to some relatively new partition statistics\, including the {\\it missi
ng excludant} (or {\\it mex}) of a partition. I will discuss unexpected c
onnections between this mex statistic and the crank\, and then we will tra
nsition to some very recent work of Blecher and Knopfmacher on partition f
ixed points which\, unbeknownst to them\, is very closely connected to the
crank and mex statistics. We will close by generalizing this concept of
partition fixed points and show how this new family of functions naturally
connects with generalized versions of the aforementioned partition statis
tics. \n\nThis is joint work with Brian Hopkins\, St. Peter's University.
\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:David and Gregory Chudnovsky (New York University)
DTSTART;VALUE=DATE-TIME:20240314T190000Z
DTEND;VALUE=DATE-TIME:20240314T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/86
DESCRIPTION:Title: The telephone gossip problem: An hommage to
Richard Bumby\nby David and Gregory Chudnovsky (New York University)
as part of New York Number Theory Seminar\n\nAbstract: TBA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin O'Bryant (College of Staten Island (CUNY))
DTSTART;VALUE=DATE-TIME:20240411T190000Z
DTEND;VALUE=DATE-TIME:20240411T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/87
DESCRIPTION:Title: $B_h$-sets\nby Kevin O'Bryant (College
of Staten Island (CUNY)) as part of New York Number Theory Seminar\n\n\nAb
stract\nFix a positive integer $h$. A $B_h$-set is a set of natural number
s that does not contain $x_i\,y_i$ with $x_1+\\cdots +x_h=y_1+\\cdots +y_h
$\, except for the trivial solutions where $x_1\,\\dots\,x_h$ is a rearran
gement of $x_1\,\\dots\,x_h$. The primary challenge is to make the $k$-th
largest element of a $B_h$-set as small as possible. This talk will contai
n the state of the art for this problem\, with special attention to how th
e problem changes as $h$ grows.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20240328T190000Z
DTEND;VALUE=DATE-TIME:20240328T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/88
DESCRIPTION:Title: Landau's converse to Holder's inequality\,
and other inequalities\nby Mel Nathanson (CUNY) as part of New York Nu
mber Theory Seminar\n\nAbstract: TBA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Leo Schaefer (University of Gottingen)
DTSTART;VALUE=DATE-TIME:20240418T190000Z
DTEND;VALUE=DATE-TIME:20240418T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/89
DESCRIPTION:Title: Telling apart coarsifications of the intege
rs\nby Leo Schaefer (University of Gottingen) as part of New York Numb
er Theory Seminar\n\n\nAbstract\nWe introduce an invariant for coarse grou
ps that is able to differentiate some coarsifications of the integers up t
o isomorphism. In particular\, we will see that coarsifications coming fro
m pro-$Q$ topologies (and therefore also the $p$-adic topologies) are not
isomorphic.\n Partial results for metrics stemming from Cayley graphs a
re also obtained\, but there remain open questions in this regard.\n \n
This is joint work with Federico Vigolo.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Borisov (Binghamton University)
DTSTART;VALUE=DATE-TIME:20240502T190000Z
DTEND;VALUE=DATE-TIME:20240502T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/90
DESCRIPTION:Title: Locally integer polynomial functions\nb
y Alexander Borisov (Binghamton University) as part of New York Number The
ory Seminar\n\n\nAbstract\nA locally integer polynomial function on a subs
et $X$ of $\\mathbb Z$ is a function $f: X\\to \\mathbb Z$ such that its
restriction to every finite subset is given by a polynomial in $\\mathbb Z
[x]$. I hope to convince you that these objects are interesting and deserv
e further study. The talk will be based on my recent preprint https://arx
iv.org/abs/2401.17955 and on further work in progress on a rather mysteri
ous analogy between locally integer polynomial functions on infinite $X$
and complex analytic functions. Several open questions will be proposed\,
highlighting how little appears to be known about these seemingly elemen
tary objects.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Quentin Dubroff (Rutgers University)
DTSTART;VALUE=DATE-TIME:20240509T190000Z
DTEND;VALUE=DATE-TIME:20240509T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/91
DESCRIPTION:Title: The Erdos distinct subset sums problem\
nby Quentin Dubroff (Rutgers University) as part of New York Number Theory
Seminar\n\n\nAbstract\nA conjecture of Erd\\H{o}s from the 1930s states
that any set of $n$ positive integers with distinct subset sums contains a
n element larger than $c2^n$ for some fixed constant c. I'll give (at leas
t) three proofs of the weaker result that any such set contains an element
larger than $c2^n/\\sqrt{n}$. Two of these proofs will achieve a bound wi
th the best-known constant $c = \\sqrt{2/\\pi}$\, which seems to be a sign
ificant sticking point. I'll highlight similarities and differences betwee
n the proofs\, which use a wide range of tools such as isoperimetric inequ
alities\, Minkowski's theorem in the geometry of numbers\, and the Berry-E
sseen quantitative central limit theorem.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Florian Luca (University of Witwatersrand)
DTSTART;VALUE=DATE-TIME:20240516T190000Z
DTEND;VALUE=DATE-TIME:20240516T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/92
DESCRIPTION:Title: On transcendence of Sturmian and Arnoux-Rau
zy words\nby Florian Luca (University of Witwatersrand) as part of New
York Number Theory Seminar\n\n\nAbstract\nWe consider numbers of the form
$\\alpha={\\displaystyle{\\sum_{n=0}^{\\infty} \\frac{u_n}{\\beta^n}}}$\,
where $(u_n)$ \nis an infinite word over a finite alphabet and $\\beta$
is a complex number of absolute\nvalue greater than one. We present a comb
inatorial criterion on $u$\, called\nechoing\, that implies that $\\alpha$
is transcendental whenever $\\beta$ is algebraic. We\nshow that every Stu
rmian word is echoing\, as is the Tribonacci word\, a leading\nexample of
an Arnoux-Rauzy word. We give an application of our\ntranscendence results
to the theory of dynamical systems\, showing that for\na contracted rotat
ion on the unit circle with algebraic slope\, its limit set is\neither fin
ite or consists exclusively of transcendental elements other than its\nend
points $0$ and $1$. This confirms a conjecture of Bugeaud\, Kim\, Laurent\
,\nand Nogueira.\n\nJoint work with P. Kebis\, A. Scoones\, J. Ouaknine an
d J. Worrell.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART;VALUE=DATE-TIME:20240905T190000Z
DTEND;VALUE=DATE-TIME:20240905T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/93
DESCRIPTION:Title: Shnirel'man density and the Dyson transform
\nby Mel Nathanson (CUNY) as part of New York Number Theory Seminar\n\
n\nAbstract\nA ``hot topic'' in the 1930s and 1940s was Khinchin's $\\alph
a+\\beta$ conjecture for the Shnirel'man density of the sum of two sets of
integers. This was solved by Henry B. Mann in 1942. The following year
Emil Artin and Peter Scherk published a refinement of his proof. In 194
5\, Freeman Dyson introduced the ``Dyson transform'' of an $n$-tuple of se
ts of positive integers and extended Mann's result to rank $r$ sums of $n$
sets of integers. The goal of this talk to simplify Dyson's method and g
eneralize his result.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Max Xu (NYU Courant)
DTSTART;VALUE=DATE-TIME:20240912T190000Z
DTEND;VALUE=DATE-TIME:20240912T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/94
DESCRIPTION:Title: Two stories about multiplicative energy
\nby Max Xu (NYU Courant) as part of New York Number Theory Seminar\n\n\nA
bstract\nThe multiplicative energy $E_{\\times}(A)$ of a given set $A$ is
defined to be the number \n of solutions to the equation \n$a_1a_2 = a_3a_
4$\,\nwhere all $a_i$ are in $A$. \n We show two recent applications of st
udying multiplicative energy. \n The first application is to study conject
ures of Elekes and Ruzsa on the size \n of product sets of arithmetic pro
gressions. \n The second story is about a recent popular topic\, random mu
ltiplicative functions\, \n and we show how multiplicative energy is invo
lved. \n The talk is based on joint work with Yunkun Zhou and K. Soundarar
ajan\, respectively.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin O'Bryant (College of Staten Island and CUNY Graduate Center)
DTSTART;VALUE=DATE-TIME:20241017T190000Z
DTEND;VALUE=DATE-TIME:20241017T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/95
DESCRIPTION:Title: Visualizing the sum-product conjecture\
nby Kevin O'Bryant (College of Staten Island and CUNY Graduate Center) as
part of New York Number Theory Seminar\n\n\nAbstract\nThe Erdos sum-produc
t conjecture states that\, for every $\\epsilon>0$\, there is $k_0$ such t
hat if $A$ is any finite set of positive integers with $|A|>k_0$\, \nthen
$|(A+A)\\cup(AA)| > |A|^{2-\\epsilon}$. In other words\, for sufficiently
large sets either the sumset or the product set will be nearly as large as
conceivable. We survey progress on this conjecture\, and provide a visual
representation of progress and counterexamples. There will be a few beaut
iful proofs (not the speaker's)\, several interesting examples\, and score
s of striking pictures.{\n
LOCATION:/talk/New_York_Number_Theory_Seminar/95/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (Lehman Callege and CUNY Graduate Center)
DTSTART;VALUE=DATE-TIME:20240919T190000Z
DTEND;VALUE=DATE-TIME:20240919T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/96
DESCRIPTION:Title: Sums of lattice points\, ordered groups\, a
nd the Hahn embedding theorem\nby Mel Nathanson (Lehman Callege and CU
NY Graduate Center) as part of New York Number Theory Seminar\n\n\nAbstrac
t\nExtension of Shnirel'man's theorem to sums of sets of nonnegative latti
ce points and to other additive problems associated with ordered groups.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (Lehman Callege and CUNY Graduate Center)
DTSTART;VALUE=DATE-TIME:20240926T190000Z
DTEND;VALUE=DATE-TIME:20240926T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/97
DESCRIPTION:Title: Addition theorems in partially ordered gro
ups\nby Mel Nathanson (Lehman Callege and CUNY Graduate Center) as par
t of New York Number Theory Seminar\n\n\nAbstract\nShnirel'man's inequalit
y and Shnirel'man's basis theorem are fundamental \nresults about sums of
sets of positive integers in additive number theory. \nIt is proved that
these results are inherently order-theoretic \nand extend to partially or
dered abelian and nonabelian groups. \nOne abelian application is an addi
tion theorem \nfor sums of sets of $n$-dimensional lattice points.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Ross (University of Hawai'i)
DTSTART;VALUE=DATE-TIME:20241031T190000Z
DTEND;VALUE=DATE-TIME:20241031T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/98
DESCRIPTION:Title: Egyptian fractions on groups\nby David
Ross (University of Hawai'i) as part of New York Number Theory Seminar\n\n
\nAbstract\nIn 1956 Sierpinski published several results about the structu
re of the set of Egyptian fractions. A few years ago Nathanson extended th
ese results to more general sets of real numbers\, and independently I sho
wed that nonstandard methods make it possible to simplify and extend Sierp
inski's results. In this talk I'll describe a further generalization\, to
certain subsets of ordered topological groups.\n
LOCATION:/talk/New_York_Number_Theory_Seminar/98/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Senia Sheydvasser (Bates College)
DTSTART;VALUE=DATE-TIME:20241114T200000Z
DTEND;VALUE=DATE-TIME:20241114T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T143534Z
UID:New_York_Number_Theory_Seminar/99
DESCRIPTION:Title: Distribution of Ulam words\nby Senia Sh
eydvasser (Bates College) as part of New York Number Theory Seminar\n\n\nA
bstract\nLet 0\,1 denote the generators of the free semigroup on two gener
ators. We say that a word is 'Ulam' if it is either 0 or 1\, or it can be
written as the concatenation of two smaller (distinct) Ulam words in exact
ly one way. This is a nonabelian analog of Ulam sequences\, defined by Bad
e et al. in 2020. In this talk\, we will discuss a few new conjectures and
results about the distribution of Ulam words---we will see that there is
a natural corresponding integer sequence\, and so it makes sense to ask qu
estions about density\, equidistribution modulo $N$\, and so on.\n
LOCATION:/talk/New_York_Number_Theory_Seminar/99/
END:VEVENT
END:VCALENDAR