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BEGIN:VEVENT
SUMMARY:Si Ying Lee (Harvard University)
DTSTART;VALUE=DATE-TIME:20200408T140000Z
DTEND;VALUE=DATE-TIME:20200408T153000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/1
DESCRIPTION:Title: Modular forms on Hilbert modular varieties\nby Si Ying Lee (Harva
rd University) as part of STAGE\n\n\nAbstract\nWe will give an overview of
Katz's paper on the construction of $p$-adic L-functions for CM fields. A
key input in this paper is the consideration of modular forms on Hilbert
modular varieties. We will discuss some key properties of Hilbert-Blumenth
al abelian varieties\, and the associated moduli spaces. We will also defi
ne modular forms on Hilbert modular varieties\, and prove a $q$-expansion
principle.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Petrov
DTSTART;VALUE=DATE-TIME:20200415T140000Z
DTEND;VALUE=DATE-TIME:20200415T153000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/2
DESCRIPTION:Title: $p$-adic modular forms on Hilbert modular varieties\nby Alexander
Petrov as part of STAGE\n\n\nAbstract\nWe will define $p$-adic Hilbert mo
dular forms via level $p^{\\infty}$ formal Hilbert modular schemes and stu
dy the relative de Rham cohomology over that scheme using the Frobenius en
domorphism.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Feng
DTSTART;VALUE=DATE-TIME:20200422T140000Z
DTEND;VALUE=DATE-TIME:20200422T153000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/3
DESCRIPTION:Title: Differential operators on modular forms\nby Tony Feng as part of
STAGE\n\n\nAbstract\nI will cover Section 2 of Katz's paper\, constructing
(analytic and p-adic) differential operators on modular forms.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ziquan Yang (Harvard University)
DTSTART;VALUE=DATE-TIME:20200429T140000Z
DTEND;VALUE=DATE-TIME:20200429T153000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/4
DESCRIPTION:Title: $p$-adic Eisenstein series\nby Ziquan Yang (Harvard University) a
s part of STAGE\n\n\nAbstract\nI will cover Section 3 of Katz's paper on $
p$-adic Eisenstein series.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zijian Yao (Harvard)
DTSTART;VALUE=DATE-TIME:20200506T140000Z
DTEND;VALUE=DATE-TIME:20200506T153000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/5
DESCRIPTION:Title: CM Hilbert-Blumenthal abelian varieties\nby Zijian Yao (Harvard)
as part of STAGE\n\n\nAbstract\nKatz 1978\, Sections 5.0 and 5.1\, and the
statements of 5.2.26 and 5.2.29.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Kriz (Massachusetts Institute of Technology)
DTSTART;VALUE=DATE-TIME:20200513T140000Z
DTEND;VALUE=DATE-TIME:20200513T153000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/6
DESCRIPTION:Title: Construction of Katz $p$-adic $L$-functions\nby Daniel Kriz (Mass
achusetts Institute of Technology) as part of STAGE\n\n\nAbstract\nWe will
describe Katz's construction of a $p$-adic measure on the $p^{\\infty}$ r
ay class group of CM fields\, whose Mellin transform is a $p$-adic $L$-fun
ction interpolating critical values of Hecke $L$-functions. First\, we wil
l recall some basics of measures and the construction of the $p$-adic modu
lar form-valued Eisenstein measure. Next\, we will obtain Katz's measure b
y evaluating the Eisenstein measure at CM points. Finally\, we will recove
r the aforementioned interpolation via Katz's insight that the values of t
he $p$-adic and complex differential operators at CM points coincide\, whi
ch follows from the moduli-theoretic definitions of these operators.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Danielle Wang (MIT)
DTSTART;VALUE=DATE-TIME:20200907T190000Z
DTEND;VALUE=DATE-TIME:20200907T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/7
DESCRIPTION:Title: Statements of the Weil conjectures\, proof for curves via the Hodge i
ndex theorem\nby Danielle Wang (MIT) as part of STAGE\n\n\nAbstract\nR
eferences: Poone
n\, Rational points on varieties\, Chapter 7 up to Section 7.5.1\; Milne\, The Riemann Hypo
thesis over Finite Fields: from Weil to the present day\, pages 8-10.\
n\nThe Weil conjectures concern the zeta functions of varieties over a fin
ite field\, which for a smooth proper variety are rational functions that
satisfy a functional equation and the Riemann hypothesis. The conjectures
led to the development of étale cohomology by Grothendieck and Artin. In
this talk\, we will state the Weil conjectures and prove the Riemann hypot
hesis for curves using the Hodge index theorem.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niven Achenjang (MIT)
DTSTART;VALUE=DATE-TIME:20200914T190000Z
DTEND;VALUE=DATE-TIME:20200914T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/8
DESCRIPTION:Title: Smooth and étale morphisms\nby Niven Achenjang (MIT) as part of
STAGE\n\n\nAbstract\nReferences: Mumford\, The red book of varieties and schemes\, III.5
and III.10\; or Poonen\, Rational points on varieties\, Section 3.5.\n\nSmooth variet
ies give an algebraic analogue of (smooth) manifolds from differential geo
metry\, while smooth and étale morphisms give algebraic analogues of subm
ersions and local isomorphisms. In addition to translating important notio
ns from differential geometry into the algebraic setting\, maps of these t
ypes play an important role in later development of étale cohomology. In
this talk\, we will introduce the definitions and basic properties of smoo
th and étale morphisms with an emphasis on providing intuition for thinki
ng about them.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthew Hase-Liu (Harvard)
DTSTART;VALUE=DATE-TIME:20200921T190000Z
DTEND;VALUE=DATE-TIME:20200921T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/9
DESCRIPTION:Title: Introduction to étale cohomology\nby Matthew Hase-Liu (Harvard)
as part of STAGE\n\n\nAbstract\nReferences: Poonen\, Rational \npoints on varieties\,
Chapter 6\; or M
ilne\, Lectures on é\;tale cohomology.\n\nA crash course on éta
le cohomology covering the following: sites and cohomology\, the étale si
te and operations on étale sheaves\, Frobenius action\, stalks of étale
sheaves\, cohomology with compact support\, and important theorems/necessi
ty of torsion coefficients.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samuel Marks (Harvard)
DTSTART;VALUE=DATE-TIME:20200928T190000Z
DTEND;VALUE=DATE-TIME:20200928T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/10
DESCRIPTION:Title: Rationality and functional equation of the zeta function\nby Sam
uel Marks (Harvard) as part of STAGE\n\n\nAbstract\nGiven a variety $X/\\m
athbb{F}_q$\, the étale cohomology groups $H^i(X_{\\overline{\\mathbb{F}_
q}}\,\\mathbb{Q}_\\ell)$ come equipped with an action of $\\mathrm{Gal}(\\
overline{\\mathbb{F}_q}/\\mathbb{F}_q)$\, and in particular with an action
of the $q$-power Frobenius. This Frobenius action can also be described a
s coming from the Frobenius morphism $\\mathrm{Fr}:X\\rightarrow X$. By us
ing these two perspectives on the Frobenius and some abstract cohomologica
l inputs\, we deduce the rationality and functional equation of $Z(X\,T)$
for nice varieties $X$.\n\nReference: Jannsen\, Deligne's proof o
f the Weil-conjecture (course notes)\, Section 1.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Roe (MIT)
DTSTART;VALUE=DATE-TIME:20201005T190000Z
DTEND;VALUE=DATE-TIME:20201005T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/11
DESCRIPTION:Title: Constructible sheaves\nby David Roe (MIT) as part of STAGE\n\n\n
Abstract\nConstructible sheaves are built from locally constant sheaves an
d serve as the coefficients for étale cohomology. We will discuss the mo
tivation behind their definition\, examples and some basic properties.\n\n
Reference: Jannsen\, Deligne's proof of the Weil-conjecture (cour
se notes)\, Section 2.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sanath Devalapurkar (Harvard)
DTSTART;VALUE=DATE-TIME:20201012T190000Z
DTEND;VALUE=DATE-TIME:20201012T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/12
DESCRIPTION:Title: Étale fundamental groups\nby Sanath Devalapurkar (Harvard) as p
art of STAGE\n\n\nAbstract\nMotivated by topological considerations\, one
can define an algebraic analogue of the fundamental group\, called the eta
le fundamental group. We will give a definition (via the abstract theory o
f Galois categories from SGA)\, and review some basic calculations.\n\nRef
erences: Milne\,
Lectures on étale cohomology\, Chapter 3\; and/or Poonen\, Rational points on variet
ies\, Sections 3.5.9 and 3.5.11.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesc Fité
DTSTART;VALUE=DATE-TIME:20201019T190000Z
DTEND;VALUE=DATE-TIME:20201019T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/13
DESCRIPTION:Title: Deligne's version of the Rankin method\nby Francesc Fité as par
t of STAGE\n\n\nAbstract\nWe will present a proof of the Riemann hypothesi
s for smooth and projective curves defined over a finite field due to Katz
. The proof reduces the general case to the case of Fermat curves via a de
formation argument (the "connect by curves lemma") and the use of Deligne'
s version of the Rankin method. For the case of Fermat curves\, we will re
call how the Riemann hypothesis amounts to a classical well-known result a
bout the size of Jacobi sums.\n\nReference: Katz\, A note on Riemann hypothesis for curves and hypersu
rfaces over finite fields\, Sections 1-4.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ziquan Yang (Harvard University)
DTSTART;VALUE=DATE-TIME:20201026T190000Z
DTEND;VALUE=DATE-TIME:20201026T203000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/14
DESCRIPTION:Title: The Riemann hypothesis for hypersurfaces\nby Ziquan Yang (Harvar
d University) as part of STAGE\n\n\nAbstract\nI will talk about Katz' meth
od of proving the Riemann hypothesis (RH) for hypersurfaces. The basic ide
a is very similar to what we saw last time: We reduce to showing RH for a
particular hypersurface. Then we show RH for this particular hypersurface
by a point-counting argument. \n\nReference: Katz\, A note on Riemann hypothesis for curves and hyper
surfaces over finite fields\, Sections 5-8.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hyuk Jun Kweon (MIT)
DTSTART;VALUE=DATE-TIME:20201102T200000Z
DTEND;VALUE=DATE-TIME:20201102T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/15
DESCRIPTION:Title: Alterations\nby Hyuk Jun Kweon (MIT) as part of STAGE\n\n\nAbstr
act\nIn 1964\, Hironaka proved that over a field of characteristic zero\,
every algebraic variety admits a resolution of singularities. However\, th
e problem of resolution of singularities is still open in positive charact
eristic. As a weaker result\, de Jong proved that every algebraic variety
admits regular alterations. We will discuss background\, main statements a
nd some applications for de Jong's result. If time allows\, we will discus
s a very rough sketch of the proof.\n\n\nReference: Notes from Conrad's le
ctures on alternations\, Section 1. The goal is to understand the sta
tement of the main theorem on alterations.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Petrov (Harvard University)
DTSTART;VALUE=DATE-TIME:20201109T200000Z
DTEND;VALUE=DATE-TIME:20201109T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/16
DESCRIPTION:Title: Weights and monodromy\nby Alexander Petrov (Harvard University)
as part of STAGE\n\n\nAbstract\nReference: Scholl\, Hypersurfaces and the Weil conjectures\, Secti
ons 1 and 2.\n\nWe will discuss the relationship between the action of loc
al monodromy around a singular fiber of a proper family and the Frobenius
action\, proving Deligne's weight monodromy theorem in equal characteristi
c.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhiyu Zhang (MIT)
DTSTART;VALUE=DATE-TIME:20201116T200000Z
DTEND;VALUE=DATE-TIME:20201116T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/17
DESCRIPTION:Title: Vanishing cycles and deformation to hypersurfaces\nby Zhiyu Zhan
g (MIT) as part of STAGE\n\n\nAbstract\nFirstly\, we give a very brief rev
iew of Weil conjecture. Following works of Scholl and Katz\, we then outli
ne a "10-line" proof of the Weil conjecture by deformation to smooth hyper
surfaces and induction on the dimension. In particular\, we will explain t
he last step i.e how to derive RH of the special fiber from the (equal cha
racteristic) weight-monodromy conjecture of the generic fiber\, using the
weight spectral sequence as an input.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raymond van Bommel (MIT)
DTSTART;VALUE=DATE-TIME:20201130T200000Z
DTEND;VALUE=DATE-TIME:20201130T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/18
DESCRIPTION:Title: The Bombieri-Stepanov approach to the Riemann hypothesis for curves
over finite fields\nby Raymond van Bommel (MIT) as part of STAGE\n\n\n
Abstract\nIn this talk\, we will discuss an elementary proof for the Riema
nn hypothesis for curves over finite fields due to Bombieri\, based on pre
vious work by Stepanov and Schmidt. It uses a method which we would now ca
ll the polynomial method\, and the Riemann Roch theorem to prove an upper
bound for the number of rational points on a curve.\n\nThe slides for the
talk will be available on Monday 30 November.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Kriz (MIT)
DTSTART;VALUE=DATE-TIME:20201207T200000Z
DTEND;VALUE=DATE-TIME:20201207T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/19
DESCRIPTION:Title: Dwork's $p$-adic proof of rationality\nby Daniel Kriz (MIT) as p
art of STAGE\n\n\nAbstract\nIn 1959\, ex-electrical engineer Bernard Dwork
shocked the mathematical world by proving the first Weil conjecture on th
e rationality of the zeta function. Dwork's proof introduced striking new
$p$-adic methods\, and defied the expectation that the Weil conjectures co
uld only be solved by developing a suitable Weil cohomology theory (later
found to be $l$-adic etale cohomology). In this talk we will outline Dwork
's proof and begin the initial part of the argument\, introducing Dwork's
general notion of "splitting functions"\, the Artin-Hasse exponential and
Dwork's lemma. \n\n\nReference: Koblitz\, p-adic numbers\, p-adic analysis\, and
zeta-functions\, pp. 92-95 and then Section V.2 to the end of the book
\, some of which may be covered in a second lecture.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Kriz (MIT)
DTSTART;VALUE=DATE-TIME:20201214T200000Z
DTEND;VALUE=DATE-TIME:20201214T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/20
DESCRIPTION:Title: Dwork's $p$-adic proof of rationality\, continued\nby Daniel Kri
z (MIT) as part of STAGE\n\n\nAbstract\nWe will go over the main steps of
Dwork's argument in detail. First\, we will construct a splitting function
for the standard additive character and show it has good convergence prop
erties using Dwork's lemma. Next we will establish the "analytic Lefschetz
fixed point formula" by studying the trace of this splitting function act
ing on $p$-adic Banach spaces of power series. Finally\, we will show this
analytic fixed point formula implies the zeta-function is the ratio of tw
o entire functions\, and conclude with a general rationality criterion for
$p$-adic power series that implies the zeta-function is rational. \n\n\nR
eference: Koblitz\, p-adic numbers\, p-adic analysis\, and zeta-functions\, w
hatever remains of Chapter V after the first lecture.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ishan Levy (MIT)
DTSTART;VALUE=DATE-TIME:20210219T180000Z
DTEND;VALUE=DATE-TIME:20210219T193000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/21
DESCRIPTION:Title: The infinitesimal site and algebraic de Rham cohomology\nby Isha
n Levy (MIT) as part of STAGE\n\n\nAbstract\nThe de Rham cohomology of the
analytification of a smooth projective\nvariety over $\\mathbb{C}$ can be
computed via an algebraic de Rham complex.\nUnfortunately\, the algebraic
de Rham complex is somewhat poorly behaved in\npositive characteristic.
To solve this problem\, Grothendieck\nshowed first how to reinterpret de R
ham cohomology in characteristic 0\nas cohomology on a site (the infinites
imal site)\, and second\nhow to modify the infinitesimal site to obtain a
site\nthat works well also in characteristic p (the crystalline site).\n\n
In this talk\, we will explain algebraic de Rham cohomology\nand define th
e infinitesimal and stratifying sites. \nWe also will define the notion of
a classical Weil cohomology theory\,\nwhich de Rham cohomology (char 0) a
nd crystalline cohomology give examples of.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Naomi Sweeting (Harvard)
DTSTART;VALUE=DATE-TIME:20210226T180000Z
DTEND;VALUE=DATE-TIME:20210226T193000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/22
DESCRIPTION:Title: Crystalline cohomology\nby Naomi Sweeting (Harvard) as part of S
TAGE\n\n\nAbstract\nThis talk will provide an overview of key concepts in
crystalline cohomology. We will begin with Grothendieck's heuristic argum
ent that\, because de Rham cohomology is independent of choice of smooth l
ift\, an intrinsic characteristic zero-valued cohomology should exist for
schemes in characteristic p. We will then discuss divided power structur
es and the crystalline site. After stating the key theorems\, we will des
cribe a relative setup in which the general theory of topoi plays a more p
rominent role. We will conclude with sketches of crucial ideas in the com
parison isomorphisms\, and a glimpse of the relationship between crystals
and connections.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Myer (The CUNY Graduate Center)
DTSTART;VALUE=DATE-TIME:20210305T180000Z
DTEND;VALUE=DATE-TIME:20210305T193000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/23
DESCRIPTION:Title: Introduction to prismatic cohomology\nby James Myer (The CUNY Gr
aduate Center) as part of STAGE\n\n\nAbstract\nThe study of the cohomology
of algebraic varieties is depicted by Peter Scholze as a “plane worth
” of pairs of primes $(p\,\\ell)$\, each indexing a cohomology theory fo
r varieties over $\\mathbb{F}_p$ with coefficients in $\\mathbb{F}_{\\ell}
$. The singular cohomology occupies a vertical line over $\\infty$\; the
étale cohomology dances around\, avoiding the pairs $(p\,p)$\; the analyt
ic de Rham cohomology occupies the top right corner\, intersecting the sin
gular cohomology @ $(\\infty\,\\infty)$\, symbolizing the classical de Rha
m comparison theorem\, while the diagonal is picked off by the algebraic d
e Rham cohomology. Zooming in on a point on the diagonal\, we begin to won
der whether there is a cohomology theory interpolating between the étale
to the crystalline (and de Rham). In fact\, the depiction of the plane of
pairs of primes is striated by lines from each of the various cohomology t
heories\, but no cohomology theory seems to “wash over” any 2-dimensio
nal part of the picture and “phase in and out” between any one or the
other. The prismatic cohomology theory is this “2-dimensional” theory
interpolating between the étale and crystalline (and de Rham) theories.\n
\nThe classical de Rham comparison theorem between the (dual of the) analy
tic de Rham cohomology and the singular homology offers a geometric interp
retation of a (co)homology class as an obstruction to (globally) integrati
ng a differential form. This geometric interpretation loses steam when fac
ed with torsion classes because the integral over a torsion class is alway
s zero. It is also worthwhile to note the relative ease with which we may
calculate the de Rham cohomology of a variety (this can be done by machine
\, e.g. Macaulay2) as opposed to the singular cohomology of a variety. So\
, how do we detect these torsion cycles algebraically? We will see via a c
alculation applying the universal coefficients theorem that the hypothesis
of equality of dimensions of the analytic and algebraic de Rham cohomolog
y groups of a variety implies lack of torsion in singular cohomology. Some
what conversely\, we’ll see that the presence of torsion in the singular
cohomology of the analytic space associated to a variety forces the algeb
raic de Rham cohomology group to be larger than expected. This interplay b
etween the various cohomology theories for varieties\, e.g. singular\, ét
ale\, analytic de Rham\, algebraic de Rham\, crystalline\, is facilitated
by a (specialization of a sequence of) remarkable theorem(s) whose proof d
epends on the existence of\, and motivates the construction of\, the prism
atic cohomology theory. \n\nFollowing this introduction\, we will venture
into some detail\, set up some notation for the next speaker\, and elabora
te a bit more on the story to come.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikayel Mkrtchyan (Harvard)
DTSTART;VALUE=DATE-TIME:20210312T180000Z
DTEND;VALUE=DATE-TIME:20210312T193000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/24
DESCRIPTION:Title: Delta rings\nby Mikayel Mkrtchyan (Harvard) as part of STAGE\n\n
\nAbstract\nThis talk will explain some basic properties of $\\delta$-ring
s\, following Bhatt-Scholze. This will include examples\, categorical prop
erties of delta-rings\, Witt vector considerations\, and\, time permitting
\, a connection with pd-envelopes.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tobias Shin (Stony Brook)
DTSTART;VALUE=DATE-TIME:20210319T170000Z
DTEND;VALUE=DATE-TIME:20210319T183000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/25
DESCRIPTION:Title: Derived categories for the working graduate student\nby Tobias S
hin (Stony Brook) as part of STAGE\n\n\nAbstract\nWe give a brief review o
f derived categories\, then discuss derived tensor products and derived co
mpletions.\n\nReferences: The Stacks project section on derived completion and the reference
s listed there.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samuel Marks (Harvard)
DTSTART;VALUE=DATE-TIME:20210326T170000Z
DTEND;VALUE=DATE-TIME:20210326T183000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/26
DESCRIPTION:Title: Distinguished elements and prisms\nby Samuel Marks (Harvard) as
part of STAGE\n\n\nAbstract\nGiven a divided power ring $(A\,I)$\, the cry
stalline site is defined using divided power thickenings over $(A\,I)$. An
alogously\, given a *prism* $(A\,I)$\, the prismatic site is defined
using "prismatic thickenings" over $(A\,I)$. The goal of this talk is to d
efine prisms and develop their basic properties.\n\nReferences: Lecture II
I of Bhatt's notes. For more details\, see the Bhatt-Scholze paper.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sanath Devalapurkar
DTSTART;VALUE=DATE-TIME:20210402T170000Z
DTEND;VALUE=DATE-TIME:20210402T183000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/27
DESCRIPTION:Title: Perfect prisms and perfectoid rings\nby Sanath Devalapurkar as p
art of STAGE\n\n\nAbstract\nWe will show that the category of perfect pris
ms is equivalent to the category of perfectoid rings\, and use this to pro
ve some structural results about perfectoid rings.\n\nReferences: Lecture
IV of Bhatt's notes. For more details\, see the Bhatt-Scholze paper.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konrad Zou (Bonn)
DTSTART;VALUE=DATE-TIME:20210409T170000Z
DTEND;VALUE=DATE-TIME:20210409T183000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/28
DESCRIPTION:Title: The prismatic site\nby Konrad Zou (Bonn) as part of STAGE\n\n\nA
bstract\nWe will introduce the prismatic site and finally define the prism
atic complex and the Hodge-Tate complex.\nWe define the Hodge-Tate compari
son map\, which relates the Kähler differentials to the cohomology of the
Hodge-Tate complex.\nFinally\, we will introduce the Čech-Alexander comp
lex\, which computes the prismatic complex in the affine case.\n\nReferenc
es: Lecture V of Bhatt's notes. For more details\, see the Bhatt-Scholze paper.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Avi Zeff (Columbia)
DTSTART;VALUE=DATE-TIME:20210416T170000Z
DTEND;VALUE=DATE-TIME:20210416T183000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/29
DESCRIPTION:Title: The Hodge-Tate and crystalline comparison theorems\nby Avi Zeff
(Columbia) as part of STAGE\n\n\nAbstract\nWe will briefly review crystall
ine cohomology and its relationship to prismatic cohomology\, and sketch a
proof of the crystalline comparison theorem and of the Hodge-Tate compari
son theorem as a corollary.\n\nReferences: Lecture VI of Bhatt's notes<
/a>. For more details\, see the Bhatt-Scholze paper.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Danielle Wang
DTSTART;VALUE=DATE-TIME:20210423T170000Z
DTEND;VALUE=DATE-TIME:20210423T183000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/30
DESCRIPTION:Title: The $q$-de Rham complex\nby Danielle Wang as part of STAGE\n\n\n
Abstract\nReferences: Lecture X of Bhatt's notes. For more details
\, see the Bhatt-Scholze paper<
/a>.\n\nIn this talk\, we define the q-de Rham complex\, show that it is a
q-deformation of the usual de Rham complex\, and state a conjecture about
the coordinate independence of this construction (to be proved next lectu
re using q-crystalline cohomology).\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Feng
DTSTART;VALUE=DATE-TIME:20210430T170000Z
DTEND;VALUE=DATE-TIME:20210430T183000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/31
DESCRIPTION:Title: $q$-crystalline cohomology\nby Tony Feng as part of STAGE\n\n\nA
bstract\nReferences: Lecture XI of Bhatt's notes. For more details
\, see the Bhatt-Scholze paper<
/a>.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zijian Yao
DTSTART;VALUE=DATE-TIME:20210507T170000Z
DTEND;VALUE=DATE-TIME:20210507T183000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/32
DESCRIPTION:Title: Extension to the singular case via derived prismatic cohomology\
nby Zijian Yao as part of STAGE\n\n\nAbstract\nReferences: Lecture VII of
Bhatt's notes. For more details\, see the Bhatt-Scholze paper.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhiyu Zhang
DTSTART;VALUE=DATE-TIME:20210514T170000Z
DTEND;VALUE=DATE-TIME:20210514T183000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/33
DESCRIPTION:Title: Perfections in mixed characteristic\nby Zhiyu Zhang as part of S
TAGE\n\n\nAbstract\nReferences: Lecture VIII of Bhatt's notes. For
more details\, see the Bhatt-S
cholze paper.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shizhang Li (University of Michigan)
DTSTART;VALUE=DATE-TIME:20210521T170000Z
DTEND;VALUE=DATE-TIME:20210521T183000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/34
DESCRIPTION:Title: The étale comparison theorem\nby Shizhang Li (University of Mic
higan) as part of STAGE\n\n\nAbstract\nReferences: Lecture IX of Bhatt'
s notes. For more details\, see the Bhatt-Scholze paper.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Angus McAndrew (Boston University)
DTSTART;VALUE=DATE-TIME:20210922T150000Z
DTEND;VALUE=DATE-TIME:20210922T163000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/35
DESCRIPTION:Title: Intersection theory with divisors\nby Angus McAndrew (Boston Uni
versity) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons
Building.\n\nAbstract\nReferences: Appendix B of Kleiman\, The Picard scheme\, Contemp. Math.\, 200
5 and/or Appendix VI.2 in Kollár\, *Rational curves on algebraic varieties
*.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nathan Chen (Harvard)
DTSTART;VALUE=DATE-TIME:20210929T140000Z
DTEND;VALUE=DATE-TIME:20210929T153000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/36
DESCRIPTION:Title: Big and nef line bundles\nby Nathan Chen (Harvard) as part of ST
AGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\
nWe will give a gentle introduction to big and nef line bundles\, with an
emphasis on their properties and examples. Reference: Sections 1.4 and 2.2
of Laz
arsfeld\, *Positivity in algebraic geometry I*\, Springer\, 2004.
\n\nNon-MIT participants must click here to get a "Tim Ticket" well in advanc
e\; this is *required* for access to the seminar.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katia Bogdanova (Harvard)
DTSTART;VALUE=DATE-TIME:20211006T150000Z
DTEND;VALUE=DATE-TIME:20211006T163000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/37
DESCRIPTION:Title: Height machine\nby Katia Bogdanova (Harvard) as part of STAGE\n\
nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nRefer
ences: Sections B1-B3 in Hindry and Silverman\, *Diophantine geometry*\, Spr
inger\, 2000 and/or Chapter 2 of Serre\, *Lectures on the Mordell-Weil theo
rem*\, 3rd edition\, Springer\, 1997.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alice Lin (Harvard)
DTSTART;VALUE=DATE-TIME:20211013T150000Z
DTEND;VALUE=DATE-TIME:20211013T163000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/38
DESCRIPTION:Title: Comparison of Weil height and canonical height\nby Alice Lin (Ha
rvard) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Bu
ilding.\n\nAbstract\nReferences: Sections B4-B5 in Hindry and Silverman\, *Diop
hantine geometry*\, Springer\, 2000 and/or Chapter 3 of Serre\, *Lecture
s on the Mordell-Weil theorem*\, 3rd edition\, Springer\, 1997. Al
so\, Theorem A of Silverman\, Heigh
ts and the specialization map for families of abelian varieties\, *J. Re
ine Angew. Math.* **342** (1983)\, 197–211.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niven Achenjang (MIT)
DTSTART;VALUE=DATE-TIME:20211020T150000Z
DTEND;VALUE=DATE-TIME:20211020T163000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/39
DESCRIPTION:Title: Vojta's approach to the Mordell conjecture I\nby Niven Achenjang
(MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Bu
ilding.\n\nAbstract\nWe will sketch Bombieri's simplification of Vojta's p
roof.\n\nReferences: Chapter 11 of Bombieri and Gubler\, *Heights in diophantine geometry*\,
Cambridge University Press\, 2006.\nand/or Part E of Hindry and Silverman\,
*Diophantine geometry*\, Springer\, 2000.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vijay Srinivasan (MIT)
DTSTART;VALUE=DATE-TIME:20211103T150000Z
DTEND;VALUE=DATE-TIME:20211103T163000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/40
DESCRIPTION:Title: Line bundles on complex tori\nby Vijay Srinivasan (MIT) as part
of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbst
ract\nSections I.1 and I.2 of Mumford\, *Abelian varieties*\, Oxford
University Press\, 1970.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Weixiao Lu (MIT)
DTSTART;VALUE=DATE-TIME:20211110T160000Z
DTEND;VALUE=DATE-TIME:20211110T173000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/41
DESCRIPTION:Title: Algebraization of complex tori\nby Weixiao Lu (MIT) as part of S
TAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract
\nSection I.3 of Mumford\, *Abelian varieties*\, Oxford University Pr
ess\, 1970.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ryan Chen (MIT)
DTSTART;VALUE=DATE-TIME:20211117T160000Z
DTEND;VALUE=DATE-TIME:20211117T173000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/42
DESCRIPTION:Title: Moduli spaces of curves and abelian varieties\nby Ryan Chen (MIT
) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Buildin
g.\nAbstract: TBA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yujie Xu (Harvard)
DTSTART;VALUE=DATE-TIME:20211201T160000Z
DTEND;VALUE=DATE-TIME:20211201T173000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/43
DESCRIPTION:Title: Betti map and Betti form I\nby Yujie Xu (Harvard) as part of STA
GE\n\nLecture held in Room 2-449 in the MIT Simons Building.\nAbstract: TB
A\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yujie Xu (Harvard)
DTSTART;VALUE=DATE-TIME:20211208T150000Z
DTEND;VALUE=DATE-TIME:20211208T163000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/45
DESCRIPTION:Title: Betti map and Betti form II\nby Yujie Xu (Harvard) as part of ST
AGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\nAbstract: T
BA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aashraya Jha (Boston University)
DTSTART;VALUE=DATE-TIME:20211215T150000Z
DTEND;VALUE=DATE-TIME:20211215T163000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/46
DESCRIPTION:Title: The height inequality and applications\nby Aashraya Jha (Boston
University) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simo
ns Building.\n\nAbstract\nWe shall look at section 7 of Ziyang Gao's summa
ry "Recent Developments of the Uniform Mordell–Lang\nConjecture". We sha
ll state the Height Inequality from the paper "Uniformity in Mordell-Lang
for curves" by Dimitrov-Gao-Habegger and an application to show a statemen
t similar to the New Gap Principle.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niven Achenjang (MIT)
DTSTART;VALUE=DATE-TIME:20211027T140000Z
DTEND;VALUE=DATE-TIME:20211027T153000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/47
DESCRIPTION:Title: Vojta's approach to the Mordell conjecture II\nby Niven Achenjan
g (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons B
uilding.\n\nAbstract\nWe will sketch Bombieri's simplification of Vojta's
proof.\n\nReferences: Chapter 11 of Bombieri and Gubler\, *Heights in diophantine geometry*\
, Cambridge University Press\, 2006.\nand/or Part E of Hindry and Silverman\,
*Diophantine geometry*\, Springer\, 2000.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Feng (MIT)
DTSTART;VALUE=DATE-TIME:20220223T150000Z
DTEND;VALUE=DATE-TIME:20220223T163000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/48
DESCRIPTION:Title: Uniform Mordell: review and preview 1\nby Tony Feng (MIT) as par
t of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\nAbst
ract: TBA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Feng (MIT)
DTSTART;VALUE=DATE-TIME:20220302T150000Z
DTEND;VALUE=DATE-TIME:20220302T163000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/49
DESCRIPTION:Title: Uniform Mordell: review and preview 2\nby Tony Feng (MIT) as par
t of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\nAbst
ract: TBA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cong Wen (Boston University)
DTSTART;VALUE=DATE-TIME:20220316T140000Z
DTEND;VALUE=DATE-TIME:20220316T153000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/50
DESCRIPTION:Title: Intersection theory and height inequality 1\nby Cong Wen (Boston
University) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Sim
ons Building.\nAbstract: TBA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xinyu Zhou (Boston University)
DTSTART;VALUE=DATE-TIME:20220330T140000Z
DTEND;VALUE=DATE-TIME:20220330T153000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/52
DESCRIPTION:Title: Intersection theory and height inequality 2\nby Xinyu Zhou (Bost
on University) as part of STAGE\n\nLecture held in Room 2-449 in the MIT S
imons Building.\nAbstract: TBA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alice Lin (Harvard)
DTSTART;VALUE=DATE-TIME:20220406T140000Z
DTEND;VALUE=DATE-TIME:20220406T153000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/53
DESCRIPTION:Title: Height bounds for nondegenerate varieties\nby Alice Lin (Harvard
) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Buildin
g.\n\nAbstract\nWe will prove the Silverman-Tate theorem in Appendix 5 of
[DGH]\, which upper-bounds the difference between the Neron-Tate height an
d the Weil height of a point $P$ in an abelian scheme $\\pi: \\mathcal{A}\
\to S$ in terms of the height of the point $\\pi(P)$ in the base scheme. T
hen\, we'll apply this result\, together with last week's Proposition 4.1
of [DGH]\, to prove Theorem 1.6 in [DGH]\, which gives a lower bound on th
e Neron-Tate height of $P$ in a nondegenerate subvariety $X$ of $\\mathcal
{A}\\to S$ in terms of the height of $\\pi(P)$. For this application\, we
follow Section 5 of [DGH].\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niven Achenjang (MIT)
DTSTART;VALUE=DATE-TIME:20220413T140000Z
DTEND;VALUE=DATE-TIME:20220413T153000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/54
DESCRIPTION:Title: Proof of the new gap principle 1\nby Niven Achenjang (MIT) as pa
rt of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nA
bstract\nOver the next two talks to prove Proposition 7.1 of [DGH] which\,
roughly-speaking\, bounds the number of points on a curve within a fixed
distance of a given point. In this talk we prepare for the proof of this p
roposition by proving a series of lemmas from section 6 of [DGH]. Specific
ally\, after stating Proposition 7.1 of [DGH]\, we will prove Theorem 6.2
(which shows non-degeneracy of a certain subvariety of the universal abeli
an variety) followed by Lemmas 6.3 and 6.1 (which will be used to obtain t
he bound in Proposition 7.1).\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aashraya Jha (Boston University)
DTSTART;VALUE=DATE-TIME:20220420T140000Z
DTEND;VALUE=DATE-TIME:20220420T153000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/55
DESCRIPTION:Title: Proof of the new gap principle 2\nby Aashraya Jha (Boston Univer
sity) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Bui
lding.\n\nAbstract\nIn this talk\, we will prove Proposition 7.1 of [DGH]\
, the so called "New Gap Principle". We will first prove a couple of lemma
s (Lemma 6.3 and Lemma 6.4 of [DGH]) using techniques from enumerative geo
metry which bounds the number of points on a given curve lying in proper s
ubsets of a certain product of varieties . We then use height bounds of po
ints on non degenerate varieties (Theorem 1.6 and Theorem 6.2 of [DGH]) al
ong with lemmas proven to use an inductive argument to prove the New Gap P
rinciple.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fei Hu (Harvard)
DTSTART;VALUE=DATE-TIME:20220427T140000Z
DTEND;VALUE=DATE-TIME:20220427T153000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/56
DESCRIPTION:Title: Uniformity for rational points\nby Fei Hu (Harvard) as part of S
TAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract
\nWe discuss the proof of Proposition 8.1 in [DGH]\, which gives a uniform
bound for the intersection of rational points $C(\\overline\\mathbb{Q})$
of a curve $C$ of large modular height in an abelian variety $A$ and a fin
ite rank subgroup $\\Gamma\\subseteq A(\\overline\\mathbb{Q})$.\nThe numbe
r of large points can be handled by a standard application of the Vojta an
d\nMumford inequalities.\nThe key of [DGH] is to bound the number of those
small points using the so-called New Gap Principle.\n\nWe then deduce the
uniform boundedness of rational/torsion points of curves in [DGH]\, i.e.\
, their Theorems 1.1\, 1.2\, and 1.4\, from the above Proposition 8.1 (for
curves of large modular height) and some other classical results (taking
care of curves of small modular height).\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anlong Chua
DTSTART;VALUE=DATE-TIME:20220504T140000Z
DTEND;VALUE=DATE-TIME:20220504T153000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/57
DESCRIPTION:Title: Unlikely intersection theory and the Ax-Schanuel theorem\nby Anl
ong Chua as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons
Building.\n\nAbstract\nCounting dimensions heuristically tells us whether
geometric objects are "likely" or "unlikely" to intersect. For instance\,
Bezout's theorem tells us that two curves in $\\mathbb{P}^2$ always inters
ect. On the other hand\, two curves in $\\mathbb{P}^3$ are unlikely to int
ersect. In number theory\, one is often concerned with unlikely intersecti
on problems — for example\, when does a subvariety of an abelian variety
contain many torsion points?\n\nIn this talk\, I will try to explain the
connections between functional transcendence\, unlikely intersections\, an
d number theory. Time permitting\, I will discuss the answer to the questi
on posed above and more. On our journey\, we will pass through the fascina
ting world of o-minimality\, which I hope to describe in broad strokes.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raymond van Bommel (Massachusetts Institute of Technology)
DTSTART;VALUE=DATE-TIME:20220511T140000Z
DTEND;VALUE=DATE-TIME:20220511T153000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/58
DESCRIPTION:Title: Proof of the amplification principle 1\nby Raymond van Bommel (M
assachusetts Institute of Technology) as part of STAGE\n\nLecture held in
Room 2-449 in the MIT Simons Building.\n\nAbstract\nWe will recall the def
initions of the Betti map and Betti rank\, and look at the degeneration lo
cus of abelian schemes. We will see how these notions are related to each
other\, and the bi-algebraic structure that we saw in the previous talk.\n
\nAll participants should abide by MIT's COVID policies https://now.mit.ed
u/policies/events/\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hyuk Jun Kweon (Massachusetts Institute of Technology)
DTSTART;VALUE=DATE-TIME:20220518T140000Z
DTEND;VALUE=DATE-TIME:20220518T153000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/59
DESCRIPTION:Title: Proof of the amplification principle 2\nby Hyuk Jun Kweon (Massa
chusetts Institute of Technology) as part of STAGE\n\nLecture held in Room
2-449 in the MIT Simons Building.\n\nAbstract\nIn the previous talk\, we
proved several results on the Betti rank. In this talk\, we will prove mor
e generalized versions of these results. Then we will prove that the rank
of Betti become maximal if we take enough iterated fibered products\, unde
r some mild conditions.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Schiavone (MIT)
DTSTART;VALUE=DATE-TIME:20220913T150000Z
DTEND;VALUE=DATE-TIME:20220913T163000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/60
DESCRIPTION:Title: Brauer groups of fields\nby Sam Schiavone (MIT) as part of STAGE
\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTo
pics: Definition of Brauer group in terms of central simple algebras (also
known as Azumaya algebras over a field)\; definition of Brauer group in t
erms of Galois cohomology\; cyclic algebras\; Brauer groups of finite fiel
ds\, local fields\, and global fields (without proofs).\n\nReferences: Poonen\, *Rationa
l \npoints on varieties*\, Section 1.5. See also Gille and Szamuel
y\, Central simple algebras and Galois cohomology\, Sections 2.4-2.6\, for
some of the topics. Also see Milne\, *Class field theory*\, Chapter IV and Theo
rem VIII.4.2.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kenta Suzuki (MIT)
DTSTART;VALUE=DATE-TIME:20220920T150000Z
DTEND;VALUE=DATE-TIME:20220920T163000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/61
DESCRIPTION:Title: Review of étale cohomology\nby Kenta Suzuki (MIT) as part of ST
AGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\
nTopic: A crash course on étale cohomology covering étale morphisms\, si
tes and cohomology\, and the étale site.\n\nReferences: Poonen\, *Rational \npoints on
varieties*\, Sections 3.5 (just enough to define étale morphism) a
nd 6.1-6.4\; or
Milne\, Lectures on é\;tale cohomology.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hao Peng (MIT)
DTSTART;VALUE=DATE-TIME:20220927T150000Z
DTEND;VALUE=DATE-TIME:20220927T163000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/62
DESCRIPTION:Title: Brauer groups of schemes\nby Hao Peng (MIT) as part of STAGE\n\n
Lecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopics
: Étale cohomology of $\\mathbb{G}_m$\; definition of cohomological Braue
r group of a scheme\; Azumaya algebras\; definition of Azumaya Brauer grou
p\; comparison (without proof).\n\nReference: Poonen\, *Rational \npoints on varieties**\, Section 6.6. See also Colliot-Thé\;lè\;ne and Skorobogatov\
, **The Brauer-Grothendieck group*\, Sections 3.1-3.3 and Chapter
4.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Haoshuo Fu (MIT)
DTSTART;VALUE=DATE-TIME:20221004T150000Z
DTEND;VALUE=DATE-TIME:20221004T163000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/63
DESCRIPTION:Title: The Hochschild-Serre spectral sequence\nby Haoshuo Fu (MIT) as p
art of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\n
Abstract\nTopics: Spectral sequences\; spectral sequence from a compositio
n of functors\; the Hochschild-Serre spectral sequence in group cohomology
and étale cohomology.\n\nReference: Poonen\, *Rational \npoints on varieties*\,
Section 6.7.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Weixiao Lu (MIT)
DTSTART;VALUE=DATE-TIME:20221011T150000Z
DTEND;VALUE=DATE-TIME:20221011T163000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/64
DESCRIPTION:Title: Residue homomorphisms and examples of Brauer groups\nby Weixiao
Lu (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons
Building.\n\nAbstract\nTopics: Residue homomorphisms\; purity\; examples o
f Brauer groups of schemes.\n\nReferences: Poonen\, *Rational \npoints on varieties*<
/a>\, Sections 6.8-6.9\; Colliot-Thé\;lè\;ne and Skorobogatov\, *Th
e Brauer-Grothendieck group*\, Section 3.7.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vijay Srinivasan (MIT)
DTSTART;VALUE=DATE-TIME:20221018T150000Z
DTEND;VALUE=DATE-TIME:20221018T163000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/65
DESCRIPTION:Title: The Brauer-Manin obstruction\nby Vijay Srinivasan (MIT) as part
of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbst
ract\nTopics: Brauer evaluation\; Brauer set\; Brauer-Manin obstruction to
the local-global principle or to weak approximation\; Brauer evaluation i
s locally constant.\n\nReference: Poonen\, *Rational \npoints on varieties*\, Sec
tions 8.2.1-8.2.4.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aashraya Jha (Boston University)
DTSTART;VALUE=DATE-TIME:20221025T150000Z
DTEND;VALUE=DATE-TIME:20221025T163000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/66
DESCRIPTION:Title: The Brauer-Manin obstruction for conic bundles\nby Aashraya Jha
(Boston University) as part of STAGE\n\nLecture held in Room 2-449 in the
MIT Simons Building.\n\nAbstract\nTopics: Iskovskikh's example\; Brauer gr
oups of conic bundles.\n\nReference: Poonen\, *Rational \npoints on varieties*\,
Section 8.2.5\; and Skorobogatov\, *Torsors and rational points*\, Se
ction 7.1.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anne Larsen (MIT)
DTSTART;VALUE=DATE-TIME:20221101T150000Z
DTEND;VALUE=DATE-TIME:20221101T163000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/67
DESCRIPTION:Title: Torsors of algebraic groups over a field\nby Anne Larsen (MIT) a
s part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\
n\nAbstract\nTopics: Torsors of groups\; torsors of algebraic groups over
a field\; examples\; classification by $H^1$\; operations on torsors.\n\nR
eference: Poonen
\, *Rational \npoints on varieties*\, Sections 5.12.1-5.12.5.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Rüd (MIT)
DTSTART;VALUE=DATE-TIME:20221108T160000Z
DTEND;VALUE=DATE-TIME:20221108T173000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/68
DESCRIPTION:Title: Torsors over finite fields\, local fields\, and global fields\nb
y Thomas Rüd (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the
MIT Simons Building.\n\nAbstract\nTopics: Torsors over fields of dimension
$\\le 1$\; torsors over local fields\; local-global principle for torsors
over global fields.\n\nReference: Poonen\, *Rational \npoints on varieties*\, Se
ctions 5.12.6-5.12.8.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alice Lin (Harvard)
DTSTART;VALUE=DATE-TIME:20221115T160000Z
DTEND;VALUE=DATE-TIME:20221115T173000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/69
DESCRIPTION:Title: Torsors over a scheme\nby Alice Lin (Harvard) as part of STAGE\n
\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopi
cs: Torsors over a scheme\; torsor sheaves\; torsors and $H^1$\; geometric
operations on torsors.\n\nReference: Poonen\, *Rational \npoints on varieties*\,
Sections 6.5.1-6.5.6.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daishi Kiyohara (MIT)
DTSTART;VALUE=DATE-TIME:20221122T160000Z
DTEND;VALUE=DATE-TIME:20221122T173000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/70
DESCRIPTION:Title: Unramified torsors\nby Daishi Kiyohara (MIT) as part of STAGE\n\
nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopic
: Unramified torsors.\n\nReference: Poonen\, *Rational \npoints on varieties*\, S
ection 6.5.7.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kush Singhal (Harvard)
DTSTART;VALUE=DATE-TIME:20221129T160000Z
DTEND;VALUE=DATE-TIME:20221129T173000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/71
DESCRIPTION:Title: An example of descent\nby Kush Singhal (Harvard) as part of STAG
E\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nT
opics: Example of descent on a genus 2 curve\; explanation in terms of twi
sts of a Galois covering.\n\nReference: Poonen\, *Rational \npoints on varieties*
\, Section 8.3.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niven Achenjang (MIT)
DTSTART;VALUE=DATE-TIME:20221206T160000Z
DTEND;VALUE=DATE-TIME:20221206T173000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/72
DESCRIPTION:Title: The descent obstruction\nby Niven Achenjang (MIT) as part of STA
GE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\n
Topics: Evaluation of torsors\; Selmer set\; weak Mordell-Weil theorem\; d
escent obstruction.\n\nReference: Poonen\, *Rational \npoints on varieties*\, Sec
tions 8.4.1-8.4.5 and 8.4.7.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xinyu Zhou (Boston University)
DTSTART;VALUE=DATE-TIME:20221213T160000Z
DTEND;VALUE=DATE-TIME:20221213T173000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/73
DESCRIPTION:Title: The étale-Brauer obstruction and insufficiency of the obstructions<
/a>\nby Xinyu Zhou (Boston University) as part of STAGE\n\nLecture held in
Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopics: The étale-Br
auer set\; comparison with the descent set\; insufficiency of the obstruct
ions for a quadric bundle over a curve.\n\nReference: Poonen\, *Rational \npoints on var
ieties*\, Sections 8.5.2-8.5.3 and 8.6.2.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kenta Suzuki (MIT)
DTSTART;VALUE=DATE-TIME:20230213T210000Z
DTEND;VALUE=DATE-TIME:20230213T223000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/74
DESCRIPTION:Title: Complex tori and abelian varieties\nby Kenta Suzuki (MIT) as par
t of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAb
stract\nWe will define abelian varieties and discuss polarization. We then
discuss Riemann's criterion for when a period matrix gives rise to an abe
lian variety\, and if we have time\, will see how the Siegel upper-half sp
ace parametrizes abelian varieties.\n\nReference: Section 1.1 (and maybe 1
.2) of Genestier a
nd Ngo\, Lectures on Shimura varieties.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Meng Yan (Brandeis University)
DTSTART;VALUE=DATE-TIME:20230227T210000Z
DTEND;VALUE=DATE-TIME:20230227T223000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/75
DESCRIPTION:Title: Quotients of the Siegel upper half space\nby Meng Yan (Brandeis
University) as part of STAGE\n\nLecture held in Room 2-135 in the MIT Simo
ns Building.\n\nAbstract\nWe will first talk about Riemann's theorem of po
larization of complex tori and then give canonical bijections between pola
rized abelian varieties and Siegel upper half-spaces. If time permits\, we
will also define principal level structures on abelian varieties to build
isomorphisms to smooth complex analytic spaces.\n\nReference: The end of
Section 1.1\, and Section 1.2 of Genestier and Ngo\, Lectures on Shimura varieties.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hao Peng (MIT)
DTSTART;VALUE=DATE-TIME:20230306T213000Z
DTEND;VALUE=DATE-TIME:20230306T230000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/76
DESCRIPTION:Title: Moduli space of abelian varieties I\nby Hao Peng (MIT) as part o
f STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstr
act\nSections 2.1-2.3 of Genestier and Ngo\, Lectures on Shimura varieties. We will firs
t finish the part on classifying isomorphism of polarized Abelian varietie
s over \\mathbb C\, then introduce dual Abelian schemes\, calculate cohomo
logy of line bundles on Abelian varieties\, and verify the representabilit
y of the moduli problem \\mathcal A classifying Abelian varieties with pol
arizations and Level strictures.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zifan Wang (MIT)
DTSTART;VALUE=DATE-TIME:20230313T203000Z
DTEND;VALUE=DATE-TIME:20230313T220000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/77
DESCRIPTION:Title: Moduli space of abelian varieties II\nby Zifan Wang (MIT) as par
t of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAb
stract\nSections 2.4.-2.6 of Genestier and Ngo\, Lectures on Shimura varieties. We will
finish the proof that the functor $\\mathcal{A}$ is represented by a smoot
h quasiprojective scheme. In particular\, to show the smoothness of $\\mat
hcal{A}$\, we review Grothendieck and Messing's theorem on deformations of
abelian schemes. Finally\, if we have time\, we will give an adelic descr
iption of $\\mathcal{A}$ and define Hecke operators.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kush Singhal (Harvard University)
DTSTART;VALUE=DATE-TIME:20230320T203000Z
DTEND;VALUE=DATE-TIME:20230320T220000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/78
DESCRIPTION:Title: Review of reductive algebraic groups\nby Kush Singhal (Harvard U
niversity) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simon
s Building.\n\nAbstract\nThis will be a crash course on the theory of redu
ctive algebraic groups. We will run through basic definitions and results
on affine algebraic groups\, reductive groups\, and tori. This will be fol
lowed by a discussion on the Lie algebra and the adjoint representation of
a reductive group. Finally\, if time allows\, we will briefly discuss Bor
el and parabolic subgroups and their relation to (generalized) flag variet
ies. No proofs will be given due to time constraints. We will mostly follo
w parts of Milne's book on Algebraic Groups (available at https://math.ucr
.edu/home/baez/qg-fall2016/Milne_iAG.pdf) specifically various subsections
of chapters 1-4\, 8\, 9\, 12\, 14\, 18\, & 19. I will thus be covering th
e prerequisites for Milne's notes on Shimura Varieties (https://www.jmilne
.org/math/xnotes/svi.pdf)\, as well as the beginning few subsections of Ch
apters 2 and 5 of these notes.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gefei Dang (MIT)
DTSTART;VALUE=DATE-TIME:20230403T203000Z
DTEND;VALUE=DATE-TIME:20230403T220000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/79
DESCRIPTION:Title: Hodge structures and variations\nby Gefei Dang (MIT) as part of
STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstrac
t\nWe will first introduce Hodge structures\, give some examples\, and rep
hrase them as representations of the Deligne torus $\\mathbb{S}$. Then we
will talk about Hodge tensors\, polarizations\, and variations of Hodge st
ructures. Finally\, we will briefly introduce hermitian symmetric domains
and realize them as parameter spaces for variations of Hodge structures.\n
\nReference: Milne\,
Introduction to Shimura varieties\, Chapter 2.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anne Larsen (MIT)
DTSTART;VALUE=DATE-TIME:20230410T203000Z
DTEND;VALUE=DATE-TIME:20230410T220000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/80
DESCRIPTION:Title: Hermitian symmetric domains and locally symmetric varieties\nby
Anne Larsen (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MI
T Simons Building.\n\nAbstract\nThe goal of this week's talk is to give th
e necessary background for the definition of a Shimura variety\, to be giv
en next week. In the first part of the talk\, we will discuss hermitian sy
mmetric domains and their groups of automorphisms (including the homomorph
ism from U_1 associated with each point and Cartan involutions on the asso
ciated real adjoint group). In the second part\, we will define arithmetic
groups and state some of the main theorems about the algebraic variety st
ructure and group of automorphisms of the quotients of hermitian symmetric
domains by torsion-free arithmetic groups.\n\nReference: Milne\, Introduction to Shimura varie
ties\, Chapter 1 and 3.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eunsu Hur (MIT)
DTSTART;VALUE=DATE-TIME:20230424T203000Z
DTEND;VALUE=DATE-TIME:20230424T220000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/81
DESCRIPTION:Title: Shimura data and Shimura varieties\nby Eunsu Hur (MIT) as part o
f STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstr
act\nPrimarily cover Chapter 4-5 of Milne.\n\nDefine congruence subgroup a
nd relate to compact open subgroups of $G(\\mathbb{A}_f)$\, no proofs nece
ssary. Define connected Shimura datum\, equivalence via Prop. 4.8. Proposi
tion 4.9. Define connected Shimura variety. Cover Example 4.14 on Hilbert
modular varieties. Give the adelic description in Prop 4.18 and Prop 4.19
.\n\nRemind us of $G^{\\mathrm{der}}$ and $G^{\\mathrm{ad}}$. Define Shimu
ra datum\, compare to connected Shimura datum. Give Ex 5.6. Cover Prop 5.7
\, Cor 5.8\, Prop 5.9. Define Shimura varieties. Define a morphism of Shim
ura varieties.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dylan Pentland (Harvard University)
DTSTART;VALUE=DATE-TIME:20230501T203000Z
DTEND;VALUE=DATE-TIME:20230501T220000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/82
DESCRIPTION:Title: Classification of Shimura varieties\nby Dylan Pentland (Harvard
University) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simo
ns Building.\n\nAbstract\nPrimarily cover Chapters 6-8 of Milne.\n\nRemind
us of the definition of a Shimura datum\, and maybe give SV2*-SV6 on p.63
. Sketch the construction of the Siegel modular variety in Chapter 6 and w
hy it satisfies SV1-SV6. Show that the Siegel modular variety parametrizes
polarized abelian varieties over $\\mathbb{C}$ with symplectic level stru
cture.\n\nSummarize Hodge type Shimura varieties as in Chapter 7.\n\nIf yo
u have time\, sketch what changes to go from Siegel modular varieties to P
EL Shimura varieties (Chapter 8). It would be great to cover some idea of
Shimura varieties of abelian type (Chapter 9).\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niven Achenjang (MIT)
DTSTART;VALUE=DATE-TIME:20230508T203000Z
DTEND;VALUE=DATE-TIME:20230508T220000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/83
DESCRIPTION:Title: Complex Multiplication\, Shimura-Taniyama formula\nby Niven Ache
njang (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simo
ns Building.\n\nAbstract\nMotivated by the desire to construct canonical m
odels of Shimura curves (in the final talk)\, we introduce the theory of c
omplex multiplication (CM) of abelian varieties. After briefly discussing
the connection between CM and canonical models\, we will cover the basic p
roperties of CM abelian varieties\, state the Shimura-Taniyama formula (wi
thout proof)\, and then give the main theorem of complex multiplication. O
ur main reference for all of this will be chapters 10 and 11 of Milne.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Landesman (MIT)
DTSTART;VALUE=DATE-TIME:20230515T203000Z
DTEND;VALUE=DATE-TIME:20230515T220000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/84
DESCRIPTION:Title: Canonical models of Shimura varieties\nby Aaron Landesman (MIT)
as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.
\n\nAbstract\nDefinition of canonical model in Chapter 12\, uniqueness in
Chapter 13\, existence in Chapter 14.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daishi Kiyohara (Harvard)
DTSTART;VALUE=DATE-TIME:20231130T210000Z
DTEND;VALUE=DATE-TIME:20231130T223000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/85
DESCRIPTION:Title: The Eisenstein quotient\nby Daishi Kiyohara (Harvard) as part of
STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstra
ct\nDefine the Eisenstein quotient and show it has the properties necessar
y to deduce the nonexistence of rational $p$-torsion in elliptic curves ov
er $\\mathbb{Q}$ for $p \\ge 11\, p\\ne 13$.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Barinder Banwait (Boston University)
DTSTART;VALUE=DATE-TIME:20231207T210000Z
DTEND;VALUE=DATE-TIME:20231207T223000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/86
DESCRIPTION:Title: Points of order 13\nby Barinder Banwait (Boston University) as p
art of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\n
Abstract\nProve that an elliptic curve over $\\mathbb{Q}$ cannot have a ra
tional point of order $13$\, following the paper of Mazur and Tate.\n\nRef
erences: [MT]\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vijay Srinivasan (MIT)
DTSTART;VALUE=DATE-TIME:20230907T200000Z
DTEND;VALUE=DATE-TIME:20230907T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/87
DESCRIPTION:Title: Overview of the proof\nby Vijay Srinivasan (MIT) as part of STAG
E\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nS
tate the main theorem and give a summary of the ingredients of the proof.\
n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhao Yu Ma (MIT)
DTSTART;VALUE=DATE-TIME:20230914T200000Z
DTEND;VALUE=DATE-TIME:20230914T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/88
DESCRIPTION:Title: Abelian varieties\nby Zhao Yu Ma (MIT) as part of STAGE\n\nLectu
re held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nBasic defin
itions regarding abelian varieties and abelian schemes (isogenies\, dual a
belian variety\, polarizations)\, Poincaré reducibility theorem\, weak Mo
rdell-Weil theorem\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alice Lin (Harvard)
DTSTART;VALUE=DATE-TIME:20230921T200000Z
DTEND;VALUE=DATE-TIME:20230921T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/89
DESCRIPTION:Title: Group schemes\nby Alice Lin (Harvard) as part of STAGE\n\nLectur
e held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nPreliminarie
s on the theory of group schemes with emphasis on finite flat group scheme
s (connected-étale sequence\, Cartier duality\, Frobenius/Verschiebung\,
Raynaud's theorem)\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Schiavone (MIT)
DTSTART;VALUE=DATE-TIME:20230928T200000Z
DTEND;VALUE=DATE-TIME:20230928T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/90
DESCRIPTION:Title: Elliptic curves over a local field\nby Sam Schiavone (MIT) as pa
rt of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nA
bstract\nBasic theory of Weierstrass equations over a DVR (including semis
table reduction theorem\, Néron-Ogg-Shafarevich criterion)\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frank Lu (Harvard)
DTSTART;VALUE=DATE-TIME:20231005T200000Z
DTEND;VALUE=DATE-TIME:20231005T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/91
DESCRIPTION:Title: Néron models\nby Frank Lu (Harvard) as part of STAGE\n\nLecture
held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nNéron models
of elliptic curves\, Néron models of abelian varieties (omitting proof o
f existence)\, reduction types of Néron models\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Hu (Harvard)
DTSTART;VALUE=DATE-TIME:20231012T200000Z
DTEND;VALUE=DATE-TIME:20231012T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/92
DESCRIPTION:Title: Relative Picard functor\nby Daniel Hu (Harvard) as part of STAGE
\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nDe
fine the relative Picard functor and discuss representability\, discuss th
e case of curves and abelian schemes.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikayel Mkrtchyan (MIT)
DTSTART;VALUE=DATE-TIME:20231019T200000Z
DTEND;VALUE=DATE-TIME:20231019T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/93
DESCRIPTION:Title: Jacobians\nby Mikayel Mkrtchyan (MIT) as part of STAGE\n\nLectur
e held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nDiscuss Jaco
bians of smooth curves\, reduced proper curves\, and families of semistabl
e curves.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eunsu Hur (MIT)
DTSTART;VALUE=DATE-TIME:20231026T200000Z
DTEND;VALUE=DATE-TIME:20231026T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/94
DESCRIPTION:Title: Modular forms and Hecke operators\nby Eunsu Hur (MIT) as part of
STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstra
ct\nTheory of modular forms and Hecke operators over $\\mathbb{C}$\, modul
ar curves over $\\mathbb{C}$ and geometric interpretation of modular forms
\, the divisor $[0]-[\\infty]$ on $X_0(p)$\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niven Achenjang (MIT)
DTSTART;VALUE=DATE-TIME:20231102T200000Z
DTEND;VALUE=DATE-TIME:20231102T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/95
DESCRIPTION:Title: Integral models of modular curves\nby Niven Achenjang (MIT) as p
art of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\n
Abstract\nSmooth models of $X_0(N)\, X_1(N)$ over $\\mathbb{Z}[1/N]$ and m
odels over $\\mathbb{Z}$ à la Deligne-Rapoport and Katz-Mazur\, consequen
ces for the structure of the Néron model of $J_0(p)$\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hari Iyer (Harvard)
DTSTART;VALUE=DATE-TIME:20231109T210000Z
DTEND;VALUE=DATE-TIME:20231109T223000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/96
DESCRIPTION:Title: Galois representations and modular forms\nby Hari Iyer (Harvard)
as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building
.\n\nAbstract\nEichler-Shimura relation on the special fiber of $J_0(N)$\,
associating Galois representations and abelian varieties to weight 2 cusp
forms.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kenta Suzuki (MIT)
DTSTART;VALUE=DATE-TIME:20231116T210000Z
DTEND;VALUE=DATE-TIME:20231116T223000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/97
DESCRIPTION:Title: Criterion for the nonexistence of rational $p$-torsion\nby Kenta
Suzuki (MIT) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Si
mons Building.\n\nAbstract\nReduce the proof of the main theorem to showin
g that there exists a rank-$0$ quotient $A$ of $J_0(p)$ such that $[0]\\ne
[\\infty]$ in $A$.\n\nReference: [Ma1] Section III.5\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raymond van Bommel (Massachusetts Institute of Technology)
DTSTART;VALUE=DATE-TIME:20240208T210000Z
DTEND;VALUE=DATE-TIME:20240208T223000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/98
DESCRIPTION:Title: Height functions\nby Raymond van Bommel (Massachusetts Institute
of Technology) as part of STAGE\n\nLecture held in Room 2-131 in the MIT
Simons Building.\n\nAbstract\nThis talk will be a survey of the theory of
heights. We will consider heights for projective varieties over number fie
lds and function fields. We will not only consider finiteness of points of
bounded degree and height\, but also the number of such points. If time a
llows\, we will consider how the heights associated to different line bund
les on a projective variety are related.\n\nThe contents of this talk are
based on Chapter 2 of the book. Another good source is Lang's book.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xinyu Zhou
DTSTART;VALUE=DATE-TIME:20240222T210000Z
DTEND;VALUE=DATE-TIME:20240222T223000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/99
DESCRIPTION:Title: Normalized heights\nby Xinyu Zhou as part of STAGE\n\nLecture he
ld in Room 2-131 in the MIT Simons Building.\n\nAbstract\nChapter 3 of Ser
re\, Lectures on the Mordell-Weil theorem\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niven T. Achenjang (MIT)
DTSTART;VALUE=DATE-TIME:20240229T210000Z
DTEND;VALUE=DATE-TIME:20240229T223000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/100
DESCRIPTION:Title: The Mordell-Weil theorem and Chabauty's theorem\nby Niven T. Ac
henjang (MIT) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Si
mons Building.\n\nAbstract\nChapter 4 and Section 5.1 of Serre\, Lectures
on the Mordell-Weil theorem.\n\nThis talk will be split into two parts. In
the first part\, we will discuss the Mordell-Weil Theorem\, which states
that the abelian group of rational points on an abelian variety $A$ define
d over a global field $K$ is finitely generated. We will show that this th
eorem follows from some classical finiteness results in algebraic number t
heory along with the theory of heights built up in previous talks. Time pe
rmitting\, we will conclude the first part by proving a theorem of Neron w
hich gives an asymptotic count for the number of points of bounded height
on an abelian variety of rank $\\rho$. In the second part\, we will turn o
ur attention towards curves of genus $g\\ge2$. For such curves $C/K$\, we
will prove Chabauty's Theorem that $C(K)$ is finite if $\\operatorname{ran
k}\\operatorname{Jac}(C)(K) < g$ (finiteness of $C(K)$ is now known even w
hen $C$'s Jacobian has large rank).\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frank Lu (Harvard)
DTSTART;VALUE=DATE-TIME:20240307T210000Z
DTEND;VALUE=DATE-TIME:20240307T223000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/101
DESCRIPTION:Title: The Demyanenko-Manin method and Mumford's inequality\nby Frank
Lu (Harvard) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Sim
ons Building.\n\nAbstract\nIn this talk\, we will discuss two theorems reg
arding the number of rational points on curves of genus $g \\geq 2:$ the D
emyanenko-Manin theorem and Mumford's inequality. We will begin with the D
emyanenko-Manin theorem\, which tells us how the existence of enough funct
ions $f_i: C \\rightarrow A\,$ for some abelian variety $A\,$ allows us to
show the number of rational points on $C$ is finite. After outlining the
proof of this theorem and discussing an application to modular curves\, we
will then sketch a proof of Mumford's inequality\, which gives an asympto
tic bound on the number of points of bounded height without knowing Faltin
g's theorem.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bjorn Poonen (MIT)
DTSTART;VALUE=DATE-TIME:20240321T200000Z
DTEND;VALUE=DATE-TIME:20240321T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/103
DESCRIPTION:Title: Siegel's method\nby Bjorn Poonen (MIT) as part of STAGE\n\nLect
ure held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nWe will sh
ow how theorems about diophantine approximation (e.g.\, Roth's theorem tha
t irrational algebraic numbers cannot be approximated too well by rational
numbers) can be used to prove one of the most famous theorems of 20th cen
tury arithmetic geometry\, Siegel's theorem that a hyperbolic affine curve
can have only finitely many integral points. The proof is ineffective\,
however: 95 years later it is still not known if there is an algorithm tha
t takes as input the equation of a curve and returns the list of its integ
ral points.\n\nReference: Chapter 7 of Serre\, Lectures on the Mordell-Wei
l theorem.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shiva Chidambaram (MIT)
DTSTART;VALUE=DATE-TIME:20240404T200000Z
DTEND;VALUE=DATE-TIME:20240404T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/104
DESCRIPTION:Title: Baker's method\nby Shiva Chidambaram (MIT) as part of STAGE\n\n
Lecture held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nIn thi
s talk\, we will discuss Baker's theorem on lower bounds for linear forms
in logarithms\, and how it gives effective bounds for quasi-integral point
s on $\\mathbb{P}^1 \\setminus \\{0\,1\,\\infty\\}$. Using coverings\, thi
s further yields effective bounds for quasi-integral points on elliptic\,
superelliptic and certain hyperelliptic affine curves. We will also discus
s an application towards finding elliptic curves with good reduction outsi
de a given finite set of places.\n\nReference: Chapter 8 of Serre\, Lectur
es on the Mordell-Weil theorem.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Hu (Harvard)
DTSTART;VALUE=DATE-TIME:20240411T200000Z
DTEND;VALUE=DATE-TIME:20240411T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/105
DESCRIPTION:Title: The Hilbert irreducibility theorem\nby Daniel Hu (Harvard) as p
art of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\n
Abstract\nI will introduce the notion of thin sets and discuss some applic
ations to Galois groups of polynomials. Then\, I will state and prove Hilb
ert's irreducibility theorem following Serre's account of Lang's proof.\n\
nReference: Chapter 9 of Serre\, Lectures on the Mordell-Weil theorem.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vijay Srinivasan
DTSTART;VALUE=DATE-TIME:20240418T200000Z
DTEND;VALUE=DATE-TIME:20240418T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/106
DESCRIPTION:Title: Construction of Galois extensions\nby Vijay Srinivasan as part
of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbst
ract\nThe inverse Galois problem asks whether every finite group can be re
alized as the Galois group of a finite extension of $\\mathbb{Q}$. In this
talk\, we will discuss the cases of $S_n$ and $\\text{PGL}_2(\\mathbb{F}_
p)$. These methods can also be adapted to apply to the simple groups $A_n$
and $\\text{PSL}_2(\\mathbb{F}_p)$ (for many $p$).\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jane Shi
DTSTART;VALUE=DATE-TIME:20240502T200000Z
DTEND;VALUE=DATE-TIME:20240502T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/107
DESCRIPTION:Title: Construction of elliptic curves of large rank\nby Jane Shi as p
art of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\n
Abstract\nIn this talk\, I will first prove Néron's theorem and explain h
ow we can use it as a basis to construct elliptic curves of large rank. Th
en\, I will discuss two methods for constructing elliptic curves of rank a
t least 9 and one method for constructing elliptic curves of rank at least
10. If there is more time\, I will discuss approaches for generating elli
ptic curves of rank at least $ 11$.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Larsen (Massachusetts Institute of Technology)
DTSTART;VALUE=DATE-TIME:20240425T200000Z
DTEND;VALUE=DATE-TIME:20240425T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/108
DESCRIPTION:Title: The large sieve\nby Daniel Larsen (Massachusetts Institute of T
echnology) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simon
s Building.\n\nAbstract\nIn this talk\, we will prove a version of the lar
ge sieve inequality\, a result from analytic number theory that will event
ually be used to give bounds on thin sets. Along the way\, we will prove t
he Davenport-Halberstam theorem and generally try to understand how the su
pport of a function's Fourier transform influences the function's behavior
.\n\nReference: Chapter 12 of Serre\, Lectures on the Mordell-Weil theorem
.\nWearing a mask is welcomed\, but optional.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hao Peng (MIT)
DTSTART;VALUE=DATE-TIME:20240509T200000Z
DTEND;VALUE=DATE-TIME:20240509T213000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/109
DESCRIPTION:Title: Applications of the large sieve to thin sets\nby Hao Peng (MIT)
as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building
.\n\nAbstract\nWe review the proof of Cohen-Serre bound on the number of r
ational points on projective and affine varieties using the large sieve me
thod and Lang-Weil bound on rational points on varieties over finite field
s. Notice that stronger bounds are known now by work of Browning\, Heath-B
rown and Salberger.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elia Gorokhovsky (Harvard)
DTSTART;VALUE=DATE-TIME:20240909T143000Z
DTEND;VALUE=DATE-TIME:20240909T160000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/110
DESCRIPTION:Title: Complex analytic spaces\, vector bundles\, and GAGA\nby Elia Go
rokhovsky (Harvard) as part of STAGE\n\nLecture held in Room 2-449 in the
MIT Simons Building.\n\nAbstract\nDefinition of the category of complex an
alytic spaces\, and statements of GAGA.\n\nReferences: \n\n- Hartsho
rne\,
*Algebraic geometry*\, 1977. Appendix B.1 and B.2. \n- G
unning and Rossi\,
*Analytic functions of several complex variables*\
, Prentice-Hall (1965). \n- Serre\, Gé\;ometrie algé\;br
ique et gé\;ometrie analytique\,
*Ann. Inst. Fourier* **6**
(1956)\, 1-42. \n- Grothendieck\, SGA I\, Exp. XII
\n

\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xinyu Fang (Harvard)
DTSTART;VALUE=DATE-TIME:20240916T143000Z
DTEND;VALUE=DATE-TIME:20240916T160000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/111
DESCRIPTION:Title: Local systems\, fundamental groupoid\, definition of connection
\nby Xinyu Fang (Harvard) as part of STAGE\n\nLecture held in Room 2-449 i
n the MIT Simons Building.\n\nAbstract\nThe definition of local systems vs
. vector bundles. Definition of the fundamental group and fundamental gro
upoid of a nice topological space. Statement of equivalence between the c
ategory of local systems and the category of finite-dimensional representa
tions of the fundamental group. Definition of connection on a vector bund
le.\n\nReference: Deligne\, up to Section 2.9.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Atticus Wang (MIT)
DTSTART;VALUE=DATE-TIME:20240923T143000Z
DTEND;VALUE=DATE-TIME:20240923T160000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/112
DESCRIPTION:Title: Integrable connections\nby Atticus Wang (MIT) as part of STAGE\
n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nDef
inition of curvature and integrable connection. Statement of the equivale
nce between the category of local systems and the category of vector bundl
es with integrable connection. Variants: schemes\, relative setting.\n\nR
eference: Deligne\, 2.10 to the end of Section 2\; Conrad\, Classical moti
vation for the Riemann-Hilbert correspondence. Notes from the talk are att
ached under "slides".\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kenta Suzuki (MIT)
DTSTART;VALUE=DATE-TIME:20240930T143000Z
DTEND;VALUE=DATE-TIME:20240930T160000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/113
DESCRIPTION:Title: Relationship to PDEs and $n$th order differential equations\nby
Kenta Suzuki (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the
MIT Simons Building.\n\nAbstract\nVia local trivializations of vector bund
les and connections\, we translate the conditions of horizontal sections a
nd flat connections in terms of classical differential equations. We then
associate vector bundles with connections to higher order differential equ
ations and finally prove an equivalence between the categories of these ob
jects (with additional data).\n\nReference: Deligne\, Sections 3 and 4.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kenz Kallal (Princeton University)
DTSTART;VALUE=DATE-TIME:20241007T143000Z
DTEND;VALUE=DATE-TIME:20241007T160000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/114
DESCRIPTION:Title: Second order differential equations and projective connections\
nby Kenz Kallal (Princeton University) as part of STAGE\n\nLecture held in
Room 2-449 in the MIT Simons Building.\n\nAbstract\nReference: Deligne\,
Section 5.\n\nIn the previous section\, Deligne sets up an equivalence of
categories between order-n differential equations on line bundles on curve
s and rank-n vector bundles with connection plus the extra data of a certa
in cyclic morphism. \n\nIn section 5\, Deligne reinterprets the special ca
se n = 2 in terms of a connection on a certain bundle and another uniformi
zation datum called a projective connection. I will prove this alternative
equivalence of categories\, focusing on the different ways of viewing and
computing with projective connections.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Barz (Princeton University)
DTSTART;VALUE=DATE-TIME:20241209T153000Z
DTEND;VALUE=DATE-TIME:20241209T170000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/115
DESCRIPTION:Title: Irregular connections and the Stokes phenomena\nby Michael Barz
(Princeton University) as part of STAGE\n\nLecture held in Room 2-449 in
the MIT Simons Building.\n\nAbstract\nDeligne's book focuses mostly on con
nections with *regular* singularities -- in the 1970s\, Deligne found
connections with irregular singularities to be pathological (see his arti
cle "Pourquoi un géomètre algébriste s'intéresse-t-il aux connexions i
rrégulières?"). But since then\, Deligne\, Malgrange\, Sibuya\, and many
others have noticed that irregular connections are home to many interesti
ng phenomena which seem to mirror things occurring for ell-adic sheaves.\n
\nRegular connections are the simplest to understand since\, by Riemann-Hi
lbert\, they are completely determined by the monodromy of their solutions
. Unfortunately\, this fails for irregular connections -- there are nontri
vial irregular connections whose solutions have no monodromy. In this talk
we describe the Stokes data which one can use to help understand irregula
r connections.\n\nReference: Malgrange\, *Équations Différentielles à
Coefficients Polynomiaux*\, chapters 3 and 4\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xinyu Zhou (Boston University)
DTSTART;VALUE=DATE-TIME:20241021T143000Z
DTEND;VALUE=DATE-TIME:20241021T160000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/116
DESCRIPTION:Title: Multivalued functions\, and regular connections in dimension 1\
nby Xinyu Zhou (Boston University) as part of STAGE\n\nLecture held in Roo
m 2-449 in the MIT Simons Building.\n\nAbstract\nDeligne\, Sections I.6 an
d the beginning of II.1.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julia Meng (MIT)
DTSTART;VALUE=DATE-TIME:20241028T143000Z
DTEND;VALUE=DATE-TIME:20241028T160000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/117
DESCRIPTION:by Julia Meng (MIT) as part of STAGE\n\nLecture held in Room 2
-449 in the MIT Simons Building.\nAbstract: TBA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:TBA
DTSTART;VALUE=DATE-TIME:20241104T153000Z
DTEND;VALUE=DATE-TIME:20241104T170000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/118
DESCRIPTION:by TBA as part of STAGE\n\nLecture held in Room 2-449 in the M
IT Simons Building.\nAbstract: TBA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:TBA
DTSTART;VALUE=DATE-TIME:20241118T153000Z
DTEND;VALUE=DATE-TIME:20241118T170000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/119
DESCRIPTION:by TBA as part of STAGE\n\nLecture held in Room 2-449 in the M
IT Simons Building.\nAbstract: TBA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:TBA
DTSTART;VALUE=DATE-TIME:20241125T153000Z
DTEND;VALUE=DATE-TIME:20241125T170000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/120
DESCRIPTION:by TBA as part of STAGE\n\nLecture held in Room 2-449 in the M
IT Simons Building.\nAbstract: TBA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:TBA
DTSTART;VALUE=DATE-TIME:20241202T153000Z
DTEND;VALUE=DATE-TIME:20241202T170000Z
DTSTAMP;VALUE=DATE-TIME:20241013T134236Z
UID:STAGE/121
DESCRIPTION:by TBA as part of STAGE\n\nLecture held in Room 2-449 in the M
IT Simons Building.\nAbstract: TBA\n
LOCATION:
END:VEVENT
END:VCALENDAR