BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Si Ying Lee (Harvard University)
DTSTART;VALUE=DATE-TIME:20200408T140000Z
DTEND;VALUE=DATE-TIME:20200408T153000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/1
DESCRIPTION:Title: Mo
dular forms on Hilbert modular varieties\nby Si Ying Lee (Harvard Univ
ersity) 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 in
put in this paper is the consideration of modular forms on Hilbert modular
varieties. We will discuss some key properties of Hilbert-Blumenthal abel
ian varieties\, and the associated moduli spaces. We will also define modu
lar forms on Hilbert modular varieties\, and prove a $q$-expansion princip
le.\n
LOCATION:https://researchseminars.org/talk/STAGE/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Petrov
DTSTART;VALUE=DATE-TIME:20200415T140000Z
DTEND;VALUE=DATE-TIME:20200415T153000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
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 modular f
orms via level $p^{\\infty}$ formal Hilbert modular schemes and study the
relative de Rham cohomology over that scheme using the Frobenius endomorph
ism.\n
LOCATION:https://researchseminars.org/talk/STAGE/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Feng
DTSTART;VALUE=DATE-TIME:20200422T140000Z
DTEND;VALUE=DATE-TIME:20200422T153000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/3
DESCRIPTION:Title: Di
fferential operators on modular forms\nby Tony Feng as part of STAGE\n
\n\nAbstract\nI will cover Section 2 of Katz's paper\, constructing (analy
tic and p-adic) differential operators on modular forms.\n
LOCATION:https://researchseminars.org/talk/STAGE/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ziquan Yang (Harvard University)
DTSTART;VALUE=DATE-TIME:20200429T140000Z
DTEND;VALUE=DATE-TIME:20200429T153000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/4
DESCRIPTION:Title: $p
$-adic Eisenstein series\nby Ziquan Yang (Harvard University) as part
of STAGE\n\n\nAbstract\nI will cover Section 3 of Katz's paper on $p$-adic
Eisenstein series.\n
LOCATION:https://researchseminars.org/talk/STAGE/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zijian Yao (Harvard)
DTSTART;VALUE=DATE-TIME:20200506T140000Z
DTEND;VALUE=DATE-TIME:20200506T153000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
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 statem
ents of 5.2.26 and 5.2.29.\n
LOCATION:https://researchseminars.org/talk/STAGE/5/
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:20240910T020501Z
UID:STAGE/6
DESCRIPTION:Title: Co
nstruction of Katz $p$-adic $L$-functions\nby Daniel Kriz (Massachuset
ts Institute of Technology) as part of STAGE\n\n\nAbstract\nWe will descri
be Katz's construction of a $p$-adic measure on the $p^{\\infty}$ ray clas
s group of CM fields\, whose Mellin transform is a $p$-adic $L$-function i
nterpolating critical values of Hecke $L$-functions. First\, we will recal
l some basics of measures and the construction of the $p$-adic modular for
m-valued Eisenstein measure. Next\, we will obtain Katz's measure by evalu
ating the Eisenstein measure at CM points. Finally\, we will recover the a
forementioned interpolation via Katz's insight that the values of the $p$-
adic and complex differential operators at CM points coincide\, which foll
ows from the moduli-theoretic definitions of these operators.\n
LOCATION:https://researchseminars.org/talk/STAGE/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Danielle Wang (MIT)
DTSTART;VALUE=DATE-TIME:20200907T190000Z
DTEND;VALUE=DATE-TIME:20200907T203000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/7
DESCRIPTION:Title: St
atements of the Weil conjectures\, proof for curves via the Hodge index th
eorem\nby Danielle Wang (MIT) as part of STAGE\n\n\nAbstract\nReferenc
es: Poonen\, Rat
ional points on varieties\, Chapter 7 up to Section 7.5.1\; Milne\, The Riemann Hypothesis
over Finite Fields: from Weil to the present day\, pages 8-10.\n\nThe
Weil conjectures concern the zeta functions of varieties over a finite fie
ld\, 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 ta
lk\, we will state the Weil conjectures and prove the Riemann hypothesis f
or curves using the Hodge index theorem.\n
LOCATION:https://researchseminars.org/talk/STAGE/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niven Achenjang (MIT)
DTSTART;VALUE=DATE-TIME:20200914T190000Z
DTEND;VALUE=DATE-TIME:20200914T203000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/8
DESCRIPTION:Title: Sm
ooth 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 II
I.10\; or Poonen
\, Rational points on varieties\, Section 3.5.\n\nSmooth varieties giv
e an algebraic analogue of (smooth) manifolds from differential geometry\,
while smooth and étale morphisms give algebraic analogues of submersions
and local isomorphisms. In addition to translating important notions from
differential geometry into the algebraic setting\, maps of these types pl
ay an important role in later development of étale cohomology. In this ta
lk\, we will introduce the definitions and basic properties of smooth and
étale morphisms with an emphasis on providing intuition for thinking abou
t them.\n
LOCATION:https://researchseminars.org/talk/STAGE/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthew Hase-Liu (Harvard)
DTSTART;VALUE=DATE-TIME:20200921T190000Z
DTEND;VALUE=DATE-TIME:20200921T203000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/9
DESCRIPTION:Title: In
troduction to étale cohomology\nby Matthew Hase-Liu (Harvard) as part
of STAGE\n\n\nAbstract\nReferences: Poonen\, Rational \npoints on varieties\, Chapter
6\; or Milne\,
Lectures on é\;tale cohomology.\n\nA crash course on étale coho
mology covering the following: sites and cohomology\, the étale site and
operations on étale sheaves\, Frobenius action\, stalks of étale sheaves
\, cohomology with compact support\, and important theorems/necessity of t
orsion coefficients.\n
LOCATION:https://researchseminars.org/talk/STAGE/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samuel Marks (Harvard)
DTSTART;VALUE=DATE-TIME:20200928T190000Z
DTEND;VALUE=DATE-TIME:20200928T203000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/10
DESCRIPTION:Title: R
ationality and functional equation of the zeta function\nby Samuel Mar
ks (Harvard) as part of STAGE\n\n\nAbstract\nGiven a variety $X/\\mathbb{F
}_q$\, the étale cohomology groups $H^i(X_{\\overline{\\mathbb{F}_q}}\,\\
mathbb{Q}_\\ell)$ come equipped with an action of $\\mathrm{Gal}(\\overlin
e{\\mathbb{F}_q}/\\mathbb{F}_q)$\, and in particular with an action of the
$q$-power Frobenius. This Frobenius action can also be described as comin
g from the Frobenius morphism $\\mathrm{Fr}:X\\rightarrow X$. By using the
se two perspectives on the Frobenius and some abstract cohomological input
s\, we deduce the rationality and functional equation of $Z(X\,T)$ for nic
e varieties $X$.\n\nReference: Jannsen\, Deligne's proof of the W
eil-conjecture (course notes)\, Section 1.\n
LOCATION:https://researchseminars.org/talk/STAGE/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Roe (MIT)
DTSTART;VALUE=DATE-TIME:20201005T190000Z
DTEND;VALUE=DATE-TIME:20201005T203000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/11
DESCRIPTION:Title: C
onstructible sheaves\nby David Roe (MIT) as part of STAGE\n\n\nAbstrac
t\nConstructible sheaves are built from locally constant sheaves and serve
as the coefficients for étale cohomology. We will discuss the motivatio
n behind their definition\, examples and some basic properties.\n\nReferen
ce: Jannsen\, Deligne's proof of the Weil-conjecture (course note
s)\, Section 2.\n
LOCATION:https://researchseminars.org/talk/STAGE/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sanath Devalapurkar (Harvard)
DTSTART;VALUE=DATE-TIME:20201012T190000Z
DTEND;VALUE=DATE-TIME:20201012T203000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/12
DESCRIPTION:Title:
Étale fundamental groups\nby Sanath Devalapurkar (Harvard) as part of
STAGE\n\n\nAbstract\nMotivated by topological considerations\, one can de
fine an algebraic analogue of the fundamental group\, called the etale fun
damental group. We will give a definition (via the abstract theory of Galo
is categories from SGA)\, and review some basic calculations.\n\nReference
s: Milne\, Lectu
res on étale cohomology\, Chapter 3\; and/or Poonen\, Rational points on varieties\, Sections 3.5.9 and 3.5.11.\n
LOCATION:https://researchseminars.org/talk/STAGE/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesc Fité
DTSTART;VALUE=DATE-TIME:20201019T190000Z
DTEND;VALUE=DATE-TIME:20201019T203000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/13
DESCRIPTION:Title: D
eligne's version of the Rankin method\nby Francesc Fité as part of ST
AGE\n\n\nAbstract\nWe will present a proof of the Riemann hypothesis for s
mooth and projective curves defined over a finite field due to Katz. The p
roof reduces the general case to the case of Fermat curves via a deformati
on argument (the "connect by curves lemma") and the use of Deligne's versi
on of the Rankin method. For the case of Fermat curves\, we will recall ho
w the Riemann hypothesis amounts to a classical well-known result about th
e size of Jacobi sums.\n\nReference: Katz\, A note on Riemann hypothesis for curves and hypersurfaces
over finite fields\, Sections 1-4.\n
LOCATION:https://researchseminars.org/talk/STAGE/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ziquan Yang (Harvard University)
DTSTART;VALUE=DATE-TIME:20201026T190000Z
DTEND;VALUE=DATE-TIME:20201026T203000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/14
DESCRIPTION:Title: T
he Riemann hypothesis for hypersurfaces\nby Ziquan Yang (Harvard Unive
rsity) as part of STAGE\n\n\nAbstract\nI will talk about Katz' method of p
roving the Riemann hypothesis (RH) for hypersurfaces. The basic idea is ve
ry similar to what we saw last time: We reduce to showing RH for a particu
lar hypersurface. Then we show RH for this particular hypersurface by a po
int-counting argument. \n\nReference: Katz\, A note on Riemann hypothesis for curves and hypersurface
s over finite fields\, Sections 5-8.\n
LOCATION:https://researchseminars.org/talk/STAGE/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hyuk Jun Kweon (MIT)
DTSTART;VALUE=DATE-TIME:20201102T200000Z
DTEND;VALUE=DATE-TIME:20201102T213000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/15
DESCRIPTION:Title: A
lterations\nby Hyuk Jun Kweon (MIT) as part of STAGE\n\n\nAbstract\nIn
1964\, Hironaka proved that over a field of characteristic zero\, every a
lgebraic variety admits a resolution of singularities. However\, the probl
em of resolution of singularities is still open in positive characteristic
. As a weaker result\, de Jong proved that every algebraic variety admits
regular alterations. We will discuss background\, main statements and some
applications for de Jong's result. If time allows\, we will discuss a ver
y rough sketch of the proof.\n\n\nReference: Notes from Conrad's lectures
on alternations\, Section 1. The goal is to understand the statement
of the main theorem on alterations.\n
LOCATION:https://researchseminars.org/talk/STAGE/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Petrov (Harvard University)
DTSTART;VALUE=DATE-TIME:20201109T200000Z
DTEND;VALUE=DATE-TIME:20201109T213000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/16
DESCRIPTION:Title: W
eights and monodromy\nby Alexander Petrov (Harvard University) as part
of STAGE\n\n\nAbstract\nReference: Scholl\, Hypersurfaces and the Weil conjectures\, Sections 1 a
nd 2.\n\nWe will discuss the relationship between the action of local mono
dromy around a singular fiber of a proper family and the Frobenius action\
, proving Deligne's weight monodromy theorem in equal characteristic.\n
LOCATION:https://researchseminars.org/talk/STAGE/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhiyu Zhang (MIT)
DTSTART;VALUE=DATE-TIME:20201116T200000Z
DTEND;VALUE=DATE-TIME:20201116T213000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/17
DESCRIPTION:Title: V
anishing cycles and deformation to hypersurfaces\nby Zhiyu Zhang (MIT)
as part of STAGE\n\n\nAbstract\nFirstly\, we give a very brief review of
Weil conjecture. Following works of Scholl and Katz\, we then outline a "1
0-line" proof of the Weil conjecture by deformation to smooth hypersurface
s and induction on the dimension. In particular\, we will explain the last
step i.e how to derive RH of the special fiber from the (equal characteri
stic) weight-monodromy conjecture of the generic fiber\, using the weight
spectral sequence as an input.\n
LOCATION:https://researchseminars.org/talk/STAGE/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raymond van Bommel (MIT)
DTSTART;VALUE=DATE-TIME:20201130T200000Z
DTEND;VALUE=DATE-TIME:20201130T213000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/18
DESCRIPTION:Title: T
he Bombieri-Stepanov approach to the Riemann hypothesis for curves over fi
nite fields\nby Raymond van Bommel (MIT) as part of STAGE\n\n\nAbstrac
t\nIn this talk\, we will discuss an elementary proof for the Riemann hypo
thesis for curves over finite fields due to Bombieri\, based on previous w
ork by Stepanov and Schmidt. It uses a method which we would now call the
polynomial method\, and the Riemann Roch theorem to prove an upper bound f
or the number of rational points on a curve.\n\nThe slides for the talk wi
ll be available on Monday 30 November.\n
LOCATION:https://researchseminars.org/talk/STAGE/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Kriz (MIT)
DTSTART;VALUE=DATE-TIME:20201207T200000Z
DTEND;VALUE=DATE-TIME:20201207T213000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/19
DESCRIPTION:Title: D
work's $p$-adic proof of rationality\nby Daniel Kriz (MIT) as part of
STAGE\n\n\nAbstract\nIn 1959\, ex-electrical engineer Bernard Dwork shocke
d the mathematical world by proving the first Weil conjecture on the ratio
nality of the zeta function. Dwork's proof introduced striking new $p$-adi
c methods\, and defied the expectation that the Weil conjectures could onl
y be solved by developing a suitable Weil cohomology theory (later found t
o be $l$-adic etale cohomology). In this talk we will outline Dwork's proo
f 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-fu
nctions\, 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:https://researchseminars.org/talk/STAGE/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Kriz (MIT)
DTSTART;VALUE=DATE-TIME:20201214T200000Z
DTEND;VALUE=DATE-TIME:20201214T213000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/20
DESCRIPTION:Title: D
work's $p$-adic proof of rationality\, continued\nby Daniel Kriz (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 th
e standard additive character and show it has good convergence properties
using Dwork's lemma. Next we will establish the "analytic Lefschetz fixed
point formula" by studying the trace of this splitting function acting on
$p$-adic Banach spaces of power series. Finally\, we will show this analyt
ic fixed point formula implies the zeta-function is the ratio of two entir
e functions\, and conclude with a general rationality criterion for $p$-ad
ic power series that implies the zeta-function is rational. \n\n\nReferenc
e: Kobl
itz\, p-adic numbers\, p-adic analysis\, and zeta-functions\, whatever
remains of Chapter V after the first lecture.\n
LOCATION:https://researchseminars.org/talk/STAGE/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ishan Levy (MIT)
DTSTART;VALUE=DATE-TIME:20210219T180000Z
DTEND;VALUE=DATE-TIME:20210219T193000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/21
DESCRIPTION:Title: T
he infinitesimal site and algebraic de Rham cohomology\nby Ishan Levy
(MIT) as part of STAGE\n\n\nAbstract\nThe de Rham cohomology of the analyt
ification of a smooth projective\nvariety over $\\mathbb{C}$ can be comput
ed via an algebraic de Rham complex.\nUnfortunately\, the algebraic de Rha
m complex is somewhat poorly behaved in\npositive characteristic. To solv
e this problem\, Grothendieck\nshowed first how to reinterpret de Rham coh
omology in characteristic 0\nas cohomology on a site (the infinitesimal si
te)\, and second\nhow to modify the infinitesimal site to obtain a site\nt
hat works well also in characteristic p (the crystalline site).\n\nIn this
talk\, we will explain algebraic de Rham cohomology\nand define the infin
itesimal and stratifying sites. \nWe also will define the notion of a clas
sical Weil cohomology theory\,\nwhich de Rham cohomology (char 0) and crys
talline cohomology give examples of.\n
LOCATION:https://researchseminars.org/talk/STAGE/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Naomi Sweeting (Harvard)
DTSTART;VALUE=DATE-TIME:20210226T180000Z
DTEND;VALUE=DATE-TIME:20210226T193000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/22
DESCRIPTION:Title: C
rystalline cohomology\nby Naomi Sweeting (Harvard) as part of STAGE\n\
n\nAbstract\nThis talk will provide an overview of key concepts in crystal
line cohomology. We will begin with Grothendieck's heuristic argument tha
t\, because de Rham cohomology is independent of choice of smooth lift\, a
n intrinsic characteristic zero-valued cohomology should exist for schemes
in characteristic p. We will then discuss divided power structures and
the crystalline site. After stating the key theorems\, we will describe a
relative setup in which the general theory of topoi plays a more prominen
t role. We will conclude with sketches of crucial ideas in the comparison
isomorphisms\, and a glimpse of the relationship between crystals and con
nections.\n
LOCATION:https://researchseminars.org/talk/STAGE/22/
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:20240910T020501Z
UID:STAGE/23
DESCRIPTION:Title: I
ntroduction to prismatic cohomology\nby James Myer (The CUNY Graduate
Center) as part of STAGE\n\n\nAbstract\nThe study of the cohomology of alg
ebraic varieties is depicted by Peter Scholze as a “plane worth” of pa
irs of primes $(p\,\\ell)$\, each indexing a cohomology theory for varieti
es over $\\mathbb{F}_p$ with coefficients in $\\mathbb{F}_{\\ell}$. The si
ngular cohomology occupies a vertical line over $\\infty$\; the étale coh
omology dances around\, avoiding the pairs $(p\,p)$\; the analytic de Rham
cohomology occupies the top right corner\, intersecting the singular coho
mology @ $(\\infty\,\\infty)$\, symbolizing the classical de Rham comparis
on theorem\, while the diagonal is picked off by the algebraic de Rham coh
omology. Zooming in on a point on the diagonal\, we begin to wonder whethe
r there is a cohomology theory interpolating between the étale to the cry
stalline (and de Rham). In fact\, the depiction of the plane of pairs of p
rimes is striated by lines from each of the various cohomology theories\,
but no cohomology theory seems to “wash over” any 2-dimensional part o
f the picture and “phase in and out” between any one or the other. The
prismatic cohomology theory is this “2-dimensional” theory interpolat
ing between the étale and crystalline (and de Rham) theories.\n\nThe clas
sical de Rham comparison theorem between the (dual of the) analytic de Rha
m cohomology and the singular homology offers a geometric interpretation o
f a (co)homology class as an obstruction to (globally) integrating a diffe
rential form. This geometric interpretation loses steam when faced with to
rsion classes because the integral over a torsion class is always 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. Ma
caulay2) as opposed to the singular cohomology of a variety. So\, how do w
e detect these torsion cycles algebraically? We will see via a calculation
applying the universal coefficients theorem that the hypothesis of equali
ty of dimensions of the analytic and algebraic de Rham cohomology groups o
f a variety implies lack of torsion in singular cohomology. Somewhat conve
rsely\, we’ll see that the presence of torsion in the singular cohomolog
y of the analytic space associated to a variety forces the algebraic de Rh
am cohomology group to be larger than expected. This interplay between the
various cohomology theories for varieties\, e.g. singular\, étale\, anal
ytic de Rham\, algebraic de Rham\, crystalline\, is facilitated by a (spec
ialization of a sequence of) remarkable theorem(s) whose proof depends on
the existence of\, and motivates the construction of\, the prismatic cohom
ology theory. \n\nFollowing this introduction\, we will venture into some
detail\, set up some notation for the next speaker\, and elaborate a bit m
ore on the story to come.\n
LOCATION:https://researchseminars.org/talk/STAGE/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikayel Mkrtchyan (Harvard)
DTSTART;VALUE=DATE-TIME:20210312T180000Z
DTEND;VALUE=DATE-TIME:20210312T193000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/24
DESCRIPTION:Title: D
elta rings\nby Mikayel Mkrtchyan (Harvard) as part of STAGE\n\n\nAbstr
act\nThis talk will explain some basic properties of $\\delta$-rings\, fol
lowing Bhatt-Scholze. This will include examples\, categorical properties
of delta-rings\, Witt vector considerations\, and\, time permitting\, a co
nnection with pd-envelopes.\n
LOCATION:https://researchseminars.org/talk/STAGE/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tobias Shin (Stony Brook)
DTSTART;VALUE=DATE-TIME:20210319T170000Z
DTEND;VALUE=DATE-TIME:20210319T183000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/25
DESCRIPTION:Title: D
erived categories for the working graduate student\nby Tobias Shin (St
ony Brook) as part of STAGE\n\n\nAbstract\nWe give a brief review of deriv
ed categories\, then discuss derived tensor products and derived completio
ns.\n\nReferences: The
Stacks project section on derived completion and the references liste
d there.\n
LOCATION:https://researchseminars.org/talk/STAGE/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samuel Marks (Harvard)
DTSTART;VALUE=DATE-TIME:20210326T170000Z
DTEND;VALUE=DATE-TIME:20210326T183000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/26
DESCRIPTION:Title: D
istinguished elements and prisms\nby Samuel Marks (Harvard) as part of
STAGE\n\n\nAbstract\nGiven a divided power ring $(A\,I)$\, the crystallin
e site is defined using divided power thickenings over $(A\,I)$. Analogous
ly\, given a *prism* $(A\,I)$\, the prismatic site is defined using "
prismatic thickenings" over $(A\,I)$. The goal of this talk is to define p
risms and develop their basic properties.\n\nReferences: Lecture III of Bhatt's notes. For more details\, see the Bhatt-Scholze paper.\n
LOCATION:https://researchseminars.org/talk/STAGE/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sanath Devalapurkar
DTSTART;VALUE=DATE-TIME:20210402T170000Z
DTEND;VALUE=DATE-TIME:20210402T183000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/27
DESCRIPTION:Title: P
erfect prisms and perfectoid rings\nby Sanath Devalapurkar as part of
STAGE\n\n\nAbstract\nWe will show that the category of perfect prisms is e
quivalent to the category of perfectoid rings\, and use this to prove some
structural results about perfectoid rings.\n\nReferences: Lecture IV of <
a href="http://www-personal.umich.edu/~bhattb/teaching/prismatic-columbia/
">Bhatt's notes. For more details\, see the Bhatt-Scholze paper.\n
LOCATION:https://researchseminars.org/talk/STAGE/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konrad Zou (Bonn)
DTSTART;VALUE=DATE-TIME:20210409T170000Z
DTEND;VALUE=DATE-TIME:20210409T183000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/28
DESCRIPTION:Title: T
he prismatic site\nby Konrad Zou (Bonn) as part of STAGE\n\n\nAbstract
\nWe will introduce the prismatic site and finally define the prismatic co
mplex and the Hodge-Tate complex.\nWe define the Hodge-Tate comparison map
\, which relates the Kähler differentials to the cohomology of the Hodge-
Tate complex.\nFinally\, we will introduce the Čech-Alexander complex\, w
hich computes the prismatic complex in the affine case.\n\nReferences: Lec
ture V of Bhatt's notes. For more details\, see the Bhatt-Scholze paper.\n
LOCATION:https://researchseminars.org/talk/STAGE/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Avi Zeff (Columbia)
DTSTART;VALUE=DATE-TIME:20210416T170000Z
DTEND;VALUE=DATE-TIME:20210416T183000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/29
DESCRIPTION:Title: T
he Hodge-Tate and crystalline comparison theorems\nby Avi Zeff (Columb
ia) as part of STAGE\n\n\nAbstract\nWe will briefly review crystalline coh
omology and its relationship to prismatic cohomology\, and sketch a proof
of the crystalline comparison theorem and of the Hodge-Tate comparison the
orem as a corollary.\n\nReferences: Lecture VI of Bhatt's notes. F
or more details\, see the Bhatt
-Scholze paper.\n
LOCATION:https://researchseminars.org/talk/STAGE/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Danielle Wang
DTSTART;VALUE=DATE-TIME:20210423T170000Z
DTEND;VALUE=DATE-TIME:20210423T183000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/30
DESCRIPTION:Title: T
he $q$-de Rham complex\nby Danielle Wang as part of STAGE\n\n\nAbstrac
t\nReferences: Lecture X of Bhatt's notes. For more details\, see
the Bhatt-Scholze paper.\n\
nIn this talk\, we define the q-de Rham complex\, show that it is a q-defo
rmation of the usual de Rham complex\, and state a conjecture about the co
ordinate independence of this construction (to be proved next lecture usin
g q-crystalline cohomology).\n
LOCATION:https://researchseminars.org/talk/STAGE/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Feng
DTSTART;VALUE=DATE-TIME:20210430T170000Z
DTEND;VALUE=DATE-TIME:20210430T183000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/31
DESCRIPTION:Title: $
q$-crystalline cohomology\nby Tony Feng as part of STAGE\n\n\nAbstract
\nReferences: Lecture XI of Bhatt's notes. For more details\, see
the Bhatt-Scholze paper.\n
LOCATION:https://researchseminars.org/talk/STAGE/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zijian Yao
DTSTART;VALUE=DATE-TIME:20210507T170000Z
DTEND;VALUE=DATE-TIME:20210507T183000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/32
DESCRIPTION:Title: E
xtension to the singular case via derived prismatic cohomology\nby Zij
ian Yao as part of STAGE\n\n\nAbstract\nReferences: Lecture VII of Bhat
t's notes. For more details\, see the Bhatt-Scholze paper.\n
LOCATION:https://researchseminars.org/talk/STAGE/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhiyu Zhang
DTSTART;VALUE=DATE-TIME:20210514T170000Z
DTEND;VALUE=DATE-TIME:20210514T183000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/33
DESCRIPTION:Title: P
erfections in mixed characteristic\nby Zhiyu Zhang as part of STAGE\n\
n\nAbstract\nReferences: Lecture VIII of Bhatt's notes. For more d
etails\, see the Bhatt-Scholze
paper.\n
LOCATION:https://researchseminars.org/talk/STAGE/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shizhang Li (University of Michigan)
DTSTART;VALUE=DATE-TIME:20210521T170000Z
DTEND;VALUE=DATE-TIME:20210521T183000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/34
DESCRIPTION:Title: T
he étale comparison theorem\nby Shizhang Li (University of Michigan)
as part of STAGE\n\n\nAbstract\nReferences: Lecture IX of Bhatt's notes
. For more details\, see the Bhatt-Scholze paper.\n
LOCATION:https://researchseminars.org/talk/STAGE/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Angus McAndrew (Boston University)
DTSTART;VALUE=DATE-TIME:20210922T150000Z
DTEND;VALUE=DATE-TIME:20210922T163000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/35
DESCRIPTION:Title: I
ntersection theory with divisors\nby Angus McAndrew (Boston University
) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Buildin
g.\n\nAbstract\nReferences: Appendix B of Kleiman\, The Picard scheme\, Contemp. Math.\, 2005 a
nd/or Appendix VI.2 in Kollár\, *Rational curves on algebraic varieties*.\n
LOCATION:https://researchseminars.org/talk/STAGE/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nathan Chen (Harvard)
DTSTART;VALUE=DATE-TIME:20210929T140000Z
DTEND;VALUE=DATE-TIME:20210929T153000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/36
DESCRIPTION:Title: B
ig and nef line bundles\nby Nathan Chen (Harvard) as part of STAGE\n\n
Lecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nWe wil
l give a gentle introduction to big and nef line bundles\, with an emphasi
s on their properties and examples. Reference: Sections 1.4 and 2.2 of Lazarsfeld
\, *Positivity in algebraic geometry I*\, Springer\, 2004.\n\nNon
-MIT participants must click here to get a "Tim Ticket" well in advance\; thi
s is *required* for access to the seminar.\n
LOCATION:https://researchseminars.org/talk/STAGE/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katia Bogdanova (Harvard)
DTSTART;VALUE=DATE-TIME:20211006T150000Z
DTEND;VALUE=DATE-TIME:20211006T163000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/37
DESCRIPTION:Title: H
eight machine\nby Katia Bogdanova (Harvard) as part of STAGE\n\nLectur
e held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nReferences:
Sections B1-B3 in Hindry and Silverman\, *Diophantine geometry*\, Springer\,
2000 and/or Chapter 2 of Serre\, *Lectures on the Mordell-Weil theorem*
\, 3rd edition\, Springer\, 1997.\n
LOCATION:https://researchseminars.org/talk/STAGE/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alice Lin (Harvard)
DTSTART;VALUE=DATE-TIME:20211013T150000Z
DTEND;VALUE=DATE-TIME:20211013T163000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/38
DESCRIPTION:Title: C
omparison of Weil height and canonical height\nby Alice Lin (Harvard)
as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.
\n\nAbstract\nReferences: Sections B4-B5 in Hindry and Silverman\, *Diophantine
geometry*\, Springer\, 2000 and/or Chapter 3 of Serre\, *Lectures on th
e Mordell-Weil theorem*\, 3rd edition\, Springer\, 1997. Also\, Th
eorem A of Silverman\, Heights and
the specialization map for families of abelian varieties\, *J. Reine Ang
ew. Math.* **342** (1983)\, 197–211.\n
LOCATION:https://researchseminars.org/talk/STAGE/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niven Achenjang (MIT)
DTSTART;VALUE=DATE-TIME:20211020T150000Z
DTEND;VALUE=DATE-TIME:20211020T163000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/39
DESCRIPTION:Title: V
ojta'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 Building.
\n\nAbstract\nWe will sketch Bombieri's simplification of Vojta's proof.\n
\nReferences: Chapter 11 of Bombieri and Gubler\, *Heights in diophantine geometry*\, Cambri
dge University Press\, 2006.\nand/or Part E of Hindry and Silverman\, *Diop
hantine geometry*\, Springer\, 2000.\n
LOCATION:https://researchseminars.org/talk/STAGE/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vijay Srinivasan (MIT)
DTSTART;VALUE=DATE-TIME:20211103T150000Z
DTEND;VALUE=DATE-TIME:20211103T163000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/40
DESCRIPTION:Title: L
ine bundles on complex tori\nby Vijay Srinivasan (MIT) as part of STAG
E\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nS
ections I.1 and I.2 of Mumford\, *Abelian varieties*\, Oxford Univers
ity Press\, 1970.\n
LOCATION:https://researchseminars.org/talk/STAGE/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Weixiao Lu (MIT)
DTSTART;VALUE=DATE-TIME:20211110T160000Z
DTEND;VALUE=DATE-TIME:20211110T173000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/41
DESCRIPTION:Title: A
lgebraization of complex tori\nby Weixiao Lu (MIT) as part of STAGE\n\
nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nSecti
on I.3 of Mumford\, *Abelian varieties*\, Oxford University Press\, 1
970.\n
LOCATION:https://researchseminars.org/talk/STAGE/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ryan Chen (MIT)
DTSTART;VALUE=DATE-TIME:20211117T160000Z
DTEND;VALUE=DATE-TIME:20211117T173000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/42
DESCRIPTION:Title: M
oduli spaces of curves and abelian varieties\nby Ryan Chen (MIT) as pa
rt of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\nAbs
tract: TBA\n
LOCATION:https://researchseminars.org/talk/STAGE/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yujie Xu (Harvard)
DTSTART;VALUE=DATE-TIME:20211201T160000Z
DTEND;VALUE=DATE-TIME:20211201T173000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/43
DESCRIPTION:Title: B
etti map and Betti form I\nby Yujie Xu (Harvard) as part of STAGE\n\nL
ecture held in Room 2-449 in the MIT Simons Building.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/STAGE/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yujie Xu (Harvard)
DTSTART;VALUE=DATE-TIME:20211208T150000Z
DTEND;VALUE=DATE-TIME:20211208T163000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/45
DESCRIPTION:Title: B
etti map and Betti form II\nby Yujie Xu (Harvard) as part of STAGE\n\n
Lecture held in Room 2-449 in the MIT Simons Building.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/STAGE/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aashraya Jha (Boston University)
DTSTART;VALUE=DATE-TIME:20211215T150000Z
DTEND;VALUE=DATE-TIME:20211215T163000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/46
DESCRIPTION:Title: T
he height inequality and applications\nby Aashraya Jha (Boston Univers
ity) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Buil
ding.\n\nAbstract\nWe shall look at section 7 of Ziyang Gao's summary "Rec
ent Developments of the Uniform Mordell–Lang\nConjecture". We shall stat
e the Height Inequality from the paper "Uniformity in Mordell-Lang for cur
ves" by Dimitrov-Gao-Habegger and an application to show a statement simil
ar to the New Gap Principle.\n
LOCATION:https://researchseminars.org/talk/STAGE/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niven Achenjang (MIT)
DTSTART;VALUE=DATE-TIME:20211027T140000Z
DTEND;VALUE=DATE-TIME:20211027T153000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/47
DESCRIPTION:Title: V
ojta's approach to the Mordell conjecture II\nby Niven Achenjang (MIT)
as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building
.\n\nAbstract\nWe will sketch Bombieri's simplification of Vojta's proof.\
n\nReferences: Chapter 11 of Bombieri and Gubler\, *Heights in diophantine geometry*\, Cambr
idge University Press\, 2006.\nand/or Part E of Hindry and Silverman\, *Dio
phantine geometry*\, Springer\, 2000.\n
LOCATION:https://researchseminars.org/talk/STAGE/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Feng (MIT)
DTSTART;VALUE=DATE-TIME:20220223T150000Z
DTEND;VALUE=DATE-TIME:20220223T163000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/48
DESCRIPTION:Title: U
niform Mordell: review and preview 1\nby Tony Feng (MIT) as part of ST
AGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\nAbstract: T
BA\n
LOCATION:https://researchseminars.org/talk/STAGE/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Feng (MIT)
DTSTART;VALUE=DATE-TIME:20220302T150000Z
DTEND;VALUE=DATE-TIME:20220302T163000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/49
DESCRIPTION:Title: U
niform Mordell: review and preview 2\nby Tony Feng (MIT) as part of ST
AGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\nAbstract: T
BA\n
LOCATION:https://researchseminars.org/talk/STAGE/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cong Wen (Boston University)
DTSTART;VALUE=DATE-TIME:20220316T140000Z
DTEND;VALUE=DATE-TIME:20220316T153000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/50
DESCRIPTION:Title: I
ntersection theory and height inequality 1\nby Cong Wen (Boston Univer
sity) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Bui
lding.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/STAGE/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xinyu Zhou (Boston University)
DTSTART;VALUE=DATE-TIME:20220330T140000Z
DTEND;VALUE=DATE-TIME:20220330T153000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/52
DESCRIPTION:Title: I
ntersection theory and height inequality 2\nby Xinyu Zhou (Boston Univ
ersity) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons B
uilding.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/STAGE/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alice Lin (Harvard)
DTSTART;VALUE=DATE-TIME:20220406T140000Z
DTEND;VALUE=DATE-TIME:20220406T153000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/53
DESCRIPTION:Title: H
eight bounds for nondegenerate varieties\nby Alice Lin (Harvard) as pa
rt of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nA
bstract\nWe will prove the Silverman-Tate theorem in Appendix 5 of [DGH]\,
which upper-bounds the difference between the Neron-Tate height and the W
eil 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. Then\, w
e'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 the 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:https://researchseminars.org/talk/STAGE/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niven Achenjang (MIT)
DTSTART;VALUE=DATE-TIME:20220413T140000Z
DTEND;VALUE=DATE-TIME:20220413T153000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/54
DESCRIPTION:Title: P
roof of the new gap principle 1\nby Niven Achenjang (MIT) as part of S
TAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract
\nOver the next two talks to prove Proposition 7.1 of [DGH] which\, roughl
y-speaking\, bounds the number of points on a curve within a fixed distanc
e of a given point. In this talk we prepare for the proof of this proposit
ion by proving a series of lemmas from section 6 of [DGH]. Specifically\,
after stating Proposition 7.1 of [DGH]\, we will prove Theorem 6.2 (which
shows non-degeneracy of a certain subvariety of the universal abelian vari
ety) followed by Lemmas 6.3 and 6.1 (which will be used to obtain the boun
d in Proposition 7.1).\n
LOCATION:https://researchseminars.org/talk/STAGE/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aashraya Jha (Boston University)
DTSTART;VALUE=DATE-TIME:20220420T140000Z
DTEND;VALUE=DATE-TIME:20220420T153000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/55
DESCRIPTION:Title: P
roof of the new gap principle 2\nby Aashraya Jha (Boston University) a
s part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\
n\nAbstract\nIn this talk\, we will prove Proposition 7.1 of [DGH]\, the s
o called "New Gap Principle". We will first prove a couple of lemmas (Lemm
a 6.3 and Lemma 6.4 of [DGH]) using techniques from enumerative geometry w
hich bounds the number of points on a given curve lying in proper subsets
of a certain product of varieties . We then use height bounds of points on
non degenerate varieties (Theorem 1.6 and Theorem 6.2 of [DGH]) along wit
h lemmas proven to use an inductive argument to prove the New Gap Principl
e.\n
LOCATION:https://researchseminars.org/talk/STAGE/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fei Hu (Harvard)
DTSTART;VALUE=DATE-TIME:20220427T140000Z
DTEND;VALUE=DATE-TIME:20220427T153000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/56
DESCRIPTION:Title: U
niformity for rational points\nby Fei Hu (Harvard) as part of STAGE\n\
nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nWe di
scuss 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 cu
rve $C$ of large modular height in an abelian variety $A$ and a finite ran
k subgroup $\\Gamma\\subseteq A(\\overline\\mathbb{Q})$.\nThe number of la
rge points can be handled by a standard application of the Vojta and\nMumf
ord 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 unifor
m 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:https://researchseminars.org/talk/STAGE/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anlong Chua
DTSTART;VALUE=DATE-TIME:20220504T140000Z
DTEND;VALUE=DATE-TIME:20220504T153000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/57
DESCRIPTION:Title: U
nlikely intersection theory and the Ax-Schanuel theorem\nby Anlong Chu
a as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Buildin
g.\n\nAbstract\nCounting dimensions heuristically tells us whether geometr
ic objects are "likely" or "unlikely" to intersect. For instance\, Bezout'
s theorem tells us that two curves in $\\mathbb{P}^2$ always intersect. On
the other hand\, two curves in $\\mathbb{P}^3$ are unlikely to intersect.
In number theory\, one is often concerned with unlikely intersection prob
lems — for example\, when does a subvariety of an abelian variety contai
n many torsion points?\n\nIn this talk\, I will try to explain the connect
ions between functional transcendence\, unlikely intersections\, and numbe
r theory. Time permitting\, I will discuss the answer to the question pose
d above and more. On our journey\, we will pass through the fascinating wo
rld of o-minimality\, which I hope to describe in broad strokes.\n
LOCATION:https://researchseminars.org/talk/STAGE/57/
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:20240910T020501Z
UID:STAGE/58
DESCRIPTION:Title: P
roof of the amplification principle 1\nby Raymond van Bommel (Massachu
setts 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 definition
s of the Betti map and Betti rank\, and look at the degeneration locus 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 p
articipants should abide by MIT's COVID policies https://now.mit.edu/polic
ies/events/\n
LOCATION:https://researchseminars.org/talk/STAGE/58/
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:20240910T020501Z
UID:STAGE/59
DESCRIPTION:Title: P
roof of the amplification principle 2\nby Hyuk Jun Kweon (Massachusett
s 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 more gener
alized versions of these results. Then we will prove that the rank of Bett
i become maximal if we take enough iterated fibered products\, under some
mild conditions.\n
LOCATION:https://researchseminars.org/talk/STAGE/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Schiavone (MIT)
DTSTART;VALUE=DATE-TIME:20220913T150000Z
DTEND;VALUE=DATE-TIME:20220913T163000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/60
DESCRIPTION:Title: B
rauer groups of fields\nby Sam Schiavone (MIT) as part of STAGE\n\nLec
ture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopics: D
efinition of Brauer group in terms of central simple algebras (also known
as Azumaya algebras over a field)\; definition of Brauer group in terms of
Galois cohomology\; cyclic algebras\; Brauer groups of finite fields\, lo
cal fields\, and global fields (without proofs).\n\nReferences: Poonen\, *Rational \npoi
nts on varieties*\, Section 1.5. See also Gille and Szamuely\, Cen
tral simple algebras and Galois cohomology\, Sections 2.4-2.6\, for some o
f the topics. Also see Milne\, *Class field theory*\, Chapter IV and Theorem VII
I.4.2.\n
LOCATION:https://researchseminars.org/talk/STAGE/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kenta Suzuki (MIT)
DTSTART;VALUE=DATE-TIME:20220920T150000Z
DTEND;VALUE=DATE-TIME:20220920T163000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/61
DESCRIPTION:Title: R
eview of étale cohomology\nby Kenta Suzuki (MIT) as part of STAGE\n\n
Lecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopic:
A crash course on étale cohomology covering étale morphisms\, sites and
cohomology\, and the étale site.\n\nReferences: Poonen\, *Rational \npoints on varieti
es*\, Sections 3.5 (just enough to define étale morphism) and 6.1-
6.4\; or Milne\,
Lectures on é\;tale cohomology.\n
LOCATION:https://researchseminars.org/talk/STAGE/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hao Peng (MIT)
DTSTART;VALUE=DATE-TIME:20220927T150000Z
DTEND;VALUE=DATE-TIME:20220927T163000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/62
DESCRIPTION:Title: B
rauer groups of schemes\nby Hao Peng (MIT) as part of STAGE\n\nLecture
held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopics: Étal
e cohomology of $\\mathbb{G}_m$\; definition of cohomological Brauer group
of a scheme\; Azumaya algebras\; definition of Azumaya Brauer group\; com
parison (without proof).\n\nReference: Poonen\, *Rational \npoints on varieties*\
, Section 6.6. See also Colliot-Thé\;lè\;ne and Skorobogatov\, *Th
e Brauer-Grothendieck group*\, Sections 3.1-3.3 and Chapter 4.\n
LOCATION:https://researchseminars.org/talk/STAGE/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Haoshuo Fu (MIT)
DTSTART;VALUE=DATE-TIME:20221004T150000Z
DTEND;VALUE=DATE-TIME:20221004T163000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/63
DESCRIPTION:Title: T
he Hochschild-Serre spectral sequence\nby Haoshuo Fu (MIT) as part of
STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstrac
t\nTopics: Spectral sequences\; spectral sequence from a composition of fu
nctors\; the Hochschild-Serre spectral sequence in group cohomology and é
tale cohomology.\n\nReference: Poonen\, *Rational \npoints on varieties*\, Sectio
n 6.7.\n
LOCATION:https://researchseminars.org/talk/STAGE/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Weixiao Lu (MIT)
DTSTART;VALUE=DATE-TIME:20221011T150000Z
DTEND;VALUE=DATE-TIME:20221011T163000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/64
DESCRIPTION:Title: R
esidue 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 Buildin
g.\n\nAbstract\nTopics: Residue homomorphisms\; purity\; examples of Braue
r groups of schemes.\n\nReferences: Poonen\, *Rational \npoints on varieties*\, S
ections 6.8-6.9\; Colliot-Thé\;lè\;ne and Skorobogatov\, *The Braue
r-Grothendieck group*\, Section 3.7.\n
LOCATION:https://researchseminars.org/talk/STAGE/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vijay Srinivasan (MIT)
DTSTART;VALUE=DATE-TIME:20221018T150000Z
DTEND;VALUE=DATE-TIME:20221018T163000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/65
DESCRIPTION:Title: T
he Brauer-Manin obstruction\nby Vijay Srinivasan (MIT) as part of STAG
E\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nT
opics: Brauer evaluation\; Brauer set\; Brauer-Manin obstruction to the lo
cal-global principle or to weak approximation\; Brauer evaluation is local
ly constant.\n\nReference: Poonen\, *Rational \npoints on varieties*\, Sections 8
.2.1-8.2.4.\n
LOCATION:https://researchseminars.org/talk/STAGE/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aashraya Jha (Boston University)
DTSTART;VALUE=DATE-TIME:20221025T150000Z
DTEND;VALUE=DATE-TIME:20221025T163000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/66
DESCRIPTION:Title: T
he 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 Sim
ons Building.\n\nAbstract\nTopics: Iskovskikh's example\; Brauer groups of
conic bundles.\n\nReference: Poonen\, *Rational \npoints on varieties*\, Section
8.2.5\; and Skorobogatov\, *Torsors and rational points*\, Section 7
.1.\n
LOCATION:https://researchseminars.org/talk/STAGE/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anne Larsen (MIT)
DTSTART;VALUE=DATE-TIME:20221101T150000Z
DTEND;VALUE=DATE-TIME:20221101T163000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/67
DESCRIPTION:Title: T
orsors of algebraic groups over a field\nby Anne Larsen (MIT) as part
of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbst
ract\nTopics: Torsors of groups\; torsors of algebraic groups over a field
\; examples\; classification by $H^1$\; operations on torsors.\n\nReferenc
e: Poonen\, *R
ational \npoints on varieties*\, Sections 5.12.1-5.12.5.\n
LOCATION:https://researchseminars.org/talk/STAGE/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Rüd (MIT)
DTSTART;VALUE=DATE-TIME:20221108T160000Z
DTEND;VALUE=DATE-TIME:20221108T173000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/68
DESCRIPTION:Title: T
orsors over finite fields\, local fields\, and global fields\nby Thoma
s Rüd (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Sim
ons Building.\n\nAbstract\nTopics: Torsors over fields of dimension $\\le
1$\; torsors over local fields\; local-global principle for torsors over g
lobal fields.\n\nReference: Poonen\, *Rational \npoints on varieties*\, Sections
5.12.6-5.12.8.\n
LOCATION:https://researchseminars.org/talk/STAGE/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alice Lin (Harvard)
DTSTART;VALUE=DATE-TIME:20221115T160000Z
DTEND;VALUE=DATE-TIME:20221115T173000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/69
DESCRIPTION:Title: T
orsors over a scheme\nby Alice Lin (Harvard) as part of STAGE\n\nLectu
re held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopics: Tor
sors over a scheme\; torsor sheaves\; torsors and $H^1$\; geometric operat
ions on torsors.\n\nReference: Poonen\, *Rational \npoints on varieties*\, Sectio
ns 6.5.1-6.5.6.\n
LOCATION:https://researchseminars.org/talk/STAGE/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daishi Kiyohara (MIT)
DTSTART;VALUE=DATE-TIME:20221122T160000Z
DTEND;VALUE=DATE-TIME:20221122T173000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/70
DESCRIPTION:Title: U
nramified torsors\nby Daishi Kiyohara (MIT) as part of STAGE\n\nLectur
e held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopic: Unram
ified torsors.\n\nReference: Poonen\, *Rational \npoints on varieties*\, Section
6.5.7.\n
LOCATION:https://researchseminars.org/talk/STAGE/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kush Singhal (Harvard)
DTSTART;VALUE=DATE-TIME:20221129T160000Z
DTEND;VALUE=DATE-TIME:20221129T173000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/71
DESCRIPTION:Title: A
n example of descent\nby Kush Singhal (Harvard) as part of STAGE\n\nLe
cture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopics:
Example of descent on a genus 2 curve\; explanation in terms of twists of
a Galois covering.\n\nReference: Poonen\, *Rational \npoints on varieties*\, Sect
ion 8.3.\n
LOCATION:https://researchseminars.org/talk/STAGE/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niven Achenjang (MIT)
DTSTART;VALUE=DATE-TIME:20221206T160000Z
DTEND;VALUE=DATE-TIME:20221206T173000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/72
DESCRIPTION:Title: T
he descent obstruction\nby Niven Achenjang (MIT) as part of STAGE\n\nL
ecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopics:
Evaluation of torsors\; Selmer set\; weak Mordell-Weil theorem\; descent
obstruction.\n\nReference: Poonen\, *Rational \npoints on varieties*\, Sections 8
.4.1-8.4.5 and 8.4.7.\n
LOCATION:https://researchseminars.org/talk/STAGE/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xinyu Zhou (Boston University)
DTSTART;VALUE=DATE-TIME:20221213T160000Z
DTEND;VALUE=DATE-TIME:20221213T173000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/73
DESCRIPTION:Title: T
he étale-Brauer obstruction and insufficiency of the obstructions\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-Brauer se
t\; comparison with the descent set\; insufficiency of the obstructions fo
r a quadric bundle over a curve.\n\nReference: Poonen\, *Rational \npoints on varieties<
/i>**\, Sections 8.5.2-8.5.3 and 8.6.2.\n
LOCATION:https://researchseminars.org/talk/STAGE/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kenta Suzuki (MIT)
DTSTART;VALUE=DATE-TIME:20230213T210000Z
DTEND;VALUE=DATE-TIME:20230213T223000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/74
DESCRIPTION:Title: C
omplex tori and abelian varieties\nby Kenta Suzuki (MIT) as part of ST
AGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\
nWe will define abelian varieties and discuss polarization. We then discus
s Riemann's criterion for when a period matrix gives rise to an abelian va
riety\, and if we have time\, will see how the Siegel upper-half space par
ametrizes abelian varieties.\n\nReference: Section 1.1 (and maybe 1.2) of
Genestier and Ngo\
, Lectures on Shimura varieties.\n
LOCATION:https://researchseminars.org/talk/STAGE/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Meng Yan (Brandeis University)
DTSTART;VALUE=DATE-TIME:20230227T210000Z
DTEND;VALUE=DATE-TIME:20230227T223000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/75
DESCRIPTION:Title: Q
uotients of the Siegel upper half space\nby Meng Yan (Brandeis Univers
ity) as part of STAGE\n\nLecture held in Room 2-135 in the MIT Simons Buil
ding.\n\nAbstract\nWe will first talk about Riemann's theorem of polarizat
ion of complex tori and then give canonical bijections between polarized a
belian varieties and Siegel upper half-spaces. If time permits\, we will a
lso define principal level structures on abelian varieties to build isomor
phisms 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:https://researchseminars.org/talk/STAGE/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hao Peng (MIT)
DTSTART;VALUE=DATE-TIME:20230306T213000Z
DTEND;VALUE=DATE-TIME:20230306T230000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/76
DESCRIPTION:Title: M
oduli space of abelian varieties I\nby Hao Peng (MIT) as part of STAGE
\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nSe
ctions 2.1-2.3 of
Genestier and Ngo\, Lectures on Shimura varieties. We will first finis
h the part on classifying isomorphism of polarized Abelian varieties over
\\mathbb C\, then introduce dual Abelian schemes\, calculate cohomology of
line bundles on Abelian varieties\, and verify the representability of th
e moduli problem \\mathcal A classifying Abelian varieties with polarizati
ons and Level strictures.\n
LOCATION:https://researchseminars.org/talk/STAGE/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zifan Wang (MIT)
DTSTART;VALUE=DATE-TIME:20230313T203000Z
DTEND;VALUE=DATE-TIME:20230313T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/77
DESCRIPTION:Title: M
oduli space of abelian varieties II\nby Zifan Wang (MIT) as part of ST
AGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\
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 smooth quasi
projective scheme. In particular\, to show the smoothness of $\\mathcal{A}
$\, we review Grothendieck and Messing's theorem on deformations of abelia
n schemes. Finally\, if we have time\, we will give an adelic description
of $\\mathcal{A}$ and define Hecke operators.\n
LOCATION:https://researchseminars.org/talk/STAGE/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kush Singhal (Harvard University)
DTSTART;VALUE=DATE-TIME:20230320T203000Z
DTEND;VALUE=DATE-TIME:20230320T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/78
DESCRIPTION:Title: R
eview of reductive algebraic groups\nby Kush Singhal (Harvard Universi
ty) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Build
ing.\n\nAbstract\nThis will be a crash course on the theory of reductive a
lgebraic groups. We will run through basic definitions and results on affi
ne algebraic groups\, reductive groups\, and tori. This will be followed b
y a discussion on the Lie algebra and the adjoint representation of a redu
ctive group. Finally\, if time allows\, we will briefly discuss Borel and
parabolic subgroups and their relation to (generalized) flag varieties. No
proofs will be given due to time constraints. We will mostly follow parts
of Milne's book on Algebraic Groups (available at https://math.ucr.edu/ho
me/baez/qg-fall2016/Milne_iAG.pdf) specifically various subsections of cha
pters 1-4\, 8\, 9\, 12\, 14\, 18\, & 19. I will thus be covering the prere
quisites for Milne's notes on Shimura Varieties (https://www.jmilne.org/ma
th/xnotes/svi.pdf)\, as well as the beginning few subsections of Chapters
2 and 5 of these notes.\n
LOCATION:https://researchseminars.org/talk/STAGE/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gefei Dang (MIT)
DTSTART;VALUE=DATE-TIME:20230403T203000Z
DTEND;VALUE=DATE-TIME:20230403T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/79
DESCRIPTION:Title: H
odge structures and variations\nby Gefei Dang (MIT) as part of STAGE\n
\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nWe w
ill first introduce Hodge structures\, give some examples\, and rephrase t
hem as representations of the Deligne torus $\\mathbb{S}$. Then we will ta
lk about Hodge tensors\, polarizations\, and variations of Hodge structure
s. Finally\, we will briefly introduce hermitian symmetric domains and rea
lize them as parameter spaces for variations of Hodge structures.\n\nRefer
ence: Milne\, Introdu
ction to Shimura varieties\, Chapter 2.\n
LOCATION:https://researchseminars.org/talk/STAGE/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anne Larsen (MIT)
DTSTART;VALUE=DATE-TIME:20230410T203000Z
DTEND;VALUE=DATE-TIME:20230410T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/80
DESCRIPTION:Title: H
ermitian symmetric domains and locally symmetric varieties\nby Anne La
rsen (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simon
s Building.\n\nAbstract\nThe goal of this week's talk is to give the neces
sary background for the definition of a Shimura variety\, to be given next
week. In the first part of the talk\, we will discuss hermitian symmetric
domains and their groups of automorphisms (including the homomorphism fro
m U_1 associated with each point and Cartan involutions on the associated
real adjoint group). In the second part\, we will define arithmetic groups
and state some of the main theorems about the algebraic variety structure
and group of automorphisms of the quotients of hermitian symmetric domain
s by torsion-free arithmetic groups.\n\nReference: Milne\, Introduction to Shimura varieties\, Chapter 1 and 3.\n
LOCATION:https://researchseminars.org/talk/STAGE/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eunsu Hur (MIT)
DTSTART;VALUE=DATE-TIME:20230424T203000Z
DTEND;VALUE=DATE-TIME:20230424T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/81
DESCRIPTION:Title: S
himura data and Shimura varieties\nby Eunsu Hur (MIT) as part of STAGE
\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nPr
imarily cover Chapter 4-5 of Milne.\n\nDefine congruence subgroup and rela
te to compact open subgroups of $G(\\mathbb{A}_f)$\, no proofs necessary.
Define connected Shimura datum\, equivalence via Prop. 4.8. Proposition 4.
9. Define connected Shimura variety. Cover Example 4.14 on Hilbert modula
r varieties. Give the adelic description in Prop 4.18 and Prop 4.19.\n\nRe
mind us of $G^{\\mathrm{der}}$ and $G^{\\mathrm{ad}}$. Define Shimura datu
m\, 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 Shimura var
ieties.\n
LOCATION:https://researchseminars.org/talk/STAGE/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dylan Pentland (Harvard University)
DTSTART;VALUE=DATE-TIME:20230501T203000Z
DTEND;VALUE=DATE-TIME:20230501T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/82
DESCRIPTION:Title: C
lassification of Shimura varieties\nby Dylan Pentland (Harvard Univers
ity) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Buil
ding.\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. Sketc
h the construction of the Siegel modular variety in Chapter 6 and why it s
atisfies SV1-SV6. Show that the Siegel modular variety parametrizes polari
zed abelian varieties over $\\mathbb{C}$ with symplectic level structure.\
n\nSummarize Hodge type Shimura varieties as in Chapter 7.\n\nIf you have
time\, sketch what changes to go from Siegel modular varieties to PEL Shim
ura varieties (Chapter 8). It would be great to cover some idea of Shimura
varieties of abelian type (Chapter 9).\n
LOCATION:https://researchseminars.org/talk/STAGE/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niven Achenjang (MIT)
DTSTART;VALUE=DATE-TIME:20230508T203000Z
DTEND;VALUE=DATE-TIME:20230508T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/83
DESCRIPTION:Title: C
omplex Multiplication\, Shimura-Taniyama formula\nby Niven Achenjang (
MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Buil
ding.\n\nAbstract\nMotivated by the desire to construct canonical models o
f Shimura curves (in the final talk)\, we introduce the theory of complex
multiplication (CM) of abelian varieties. After briefly discussing the con
nection between CM and canonical models\, we will cover the basic properti
es of CM abelian varieties\, state the Shimura-Taniyama formula (without p
roof)\, and then give the main theorem of complex multiplication. Our main
reference for all of this will be chapters 10 and 11 of Milne.\n
LOCATION:https://researchseminars.org/talk/STAGE/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Landesman (MIT)
DTSTART;VALUE=DATE-TIME:20230515T203000Z
DTEND;VALUE=DATE-TIME:20230515T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/84
DESCRIPTION:Title: C
anonical 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\nAbs
tract\nDefinition of canonical model in Chapter 12\, uniqueness in Chapter
13\, existence in Chapter 14.\n
LOCATION:https://researchseminars.org/talk/STAGE/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daishi Kiyohara (Harvard)
DTSTART;VALUE=DATE-TIME:20231130T210000Z
DTEND;VALUE=DATE-TIME:20231130T223000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/85
DESCRIPTION:Title: T
he Eisenstein quotient\nby Daishi Kiyohara (Harvard) as part of STAGE\
n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nDef
ine the Eisenstein quotient and show it has the properties necessary to de
duce the nonexistence of rational $p$-torsion in elliptic curves over $\\m
athbb{Q}$ for $p \\ge 11\, p\\ne 13$.\n
LOCATION:https://researchseminars.org/talk/STAGE/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Barinder Banwait (Boston University)
DTSTART;VALUE=DATE-TIME:20231207T210000Z
DTEND;VALUE=DATE-TIME:20231207T223000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/86
DESCRIPTION:Title: P
oints of order 13\nby Barinder Banwait (Boston University) as part of
STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstrac
t\nProve that an elliptic curve over $\\mathbb{Q}$ cannot have a rational
point of order $13$\, following the paper of Mazur and Tate.\n\nReferences
: [MT]\n
LOCATION:https://researchseminars.org/talk/STAGE/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vijay Srinivasan (MIT)
DTSTART;VALUE=DATE-TIME:20230907T200000Z
DTEND;VALUE=DATE-TIME:20230907T213000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/87
DESCRIPTION:Title: O
verview of the proof\nby Vijay Srinivasan (MIT) as part of STAGE\n\nLe
cture held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nState th
e main theorem and give a summary of the ingredients of the proof.\n
LOCATION:https://researchseminars.org/talk/STAGE/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhao Yu Ma (MIT)
DTSTART;VALUE=DATE-TIME:20230914T200000Z
DTEND;VALUE=DATE-TIME:20230914T213000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/88
DESCRIPTION:Title: A
belian varieties\nby Zhao Yu Ma (MIT) as part of STAGE\n\nLecture held
in Room 2-131 in the MIT Simons Building.\n\nAbstract\nBasic definitions
regarding abelian varieties and abelian schemes (isogenies\, dual abelian
variety\, polarizations)\, Poincaré reducibility theorem\, weak Mordell-W
eil theorem\n
LOCATION:https://researchseminars.org/talk/STAGE/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alice Lin (Harvard)
DTSTART;VALUE=DATE-TIME:20230921T200000Z
DTEND;VALUE=DATE-TIME:20230921T213000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/89
DESCRIPTION:Title: G
roup schemes\nby Alice Lin (Harvard) as part of STAGE\n\nLecture held
in Room 2-131 in the MIT Simons Building.\n\nAbstract\nPreliminaries on th
e theory of group schemes with emphasis on finite flat group schemes (conn
ected-étale sequence\, Cartier duality\, Frobenius/Verschiebung\, Raynaud
's theorem)\n
LOCATION:https://researchseminars.org/talk/STAGE/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Schiavone (MIT)
DTSTART;VALUE=DATE-TIME:20230928T200000Z
DTEND;VALUE=DATE-TIME:20230928T213000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/90
DESCRIPTION:Title: E
lliptic curves over a local field\nby Sam Schiavone (MIT) as part of S
TAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstract
\nBasic theory of Weierstrass equations over a DVR (including semistable r
eduction theorem\, Néron-Ogg-Shafarevich criterion)\n
LOCATION:https://researchseminars.org/talk/STAGE/90/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frank Lu (Harvard)
DTSTART;VALUE=DATE-TIME:20231005T200000Z
DTEND;VALUE=DATE-TIME:20231005T213000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/91
DESCRIPTION:Title: N
éron models\nby Frank Lu (Harvard) as part of STAGE\n\nLecture held i
n Room 2-131 in the MIT Simons Building.\n\nAbstract\nNéron models of ell
iptic curves\, Néron models of abelian varieties (omitting proof of exist
ence)\, reduction types of Néron models\n
LOCATION:https://researchseminars.org/talk/STAGE/91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Hu (Harvard)
DTSTART;VALUE=DATE-TIME:20231012T200000Z
DTEND;VALUE=DATE-TIME:20231012T213000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/92
DESCRIPTION:Title: R
elative Picard functor\nby Daniel Hu (Harvard) as part of STAGE\n\nLec
ture held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nDefine th
e relative Picard functor and discuss representability\, discuss the case
of curves and abelian schemes.\n
LOCATION:https://researchseminars.org/talk/STAGE/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikayel Mkrtchyan (MIT)
DTSTART;VALUE=DATE-TIME:20231019T200000Z
DTEND;VALUE=DATE-TIME:20231019T213000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/93
DESCRIPTION:Title: J
acobians\nby Mikayel Mkrtchyan (MIT) as part of STAGE\n\nLecture held
in Room 2-131 in the MIT Simons Building.\n\nAbstract\nDiscuss Jacobians o
f smooth curves\, reduced proper curves\, and families of semistable curve
s.\n
LOCATION:https://researchseminars.org/talk/STAGE/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eunsu Hur (MIT)
DTSTART;VALUE=DATE-TIME:20231026T200000Z
DTEND;VALUE=DATE-TIME:20231026T213000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/94
DESCRIPTION:Title: M
odular 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\nAbstract\nThe
ory of modular forms and Hecke operators over $\\mathbb{C}$\, modular curv
es over $\\mathbb{C}$ and geometric interpretation of modular forms\, the
divisor $[0]-[\\infty]$ on $X_0(p)$\n
LOCATION:https://researchseminars.org/talk/STAGE/94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niven Achenjang (MIT)
DTSTART;VALUE=DATE-TIME:20231102T200000Z
DTEND;VALUE=DATE-TIME:20231102T213000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/95
DESCRIPTION:Title: I
ntegral models of modular curves\nby Niven Achenjang (MIT) as part of
STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstrac
t\nSmooth models of $X_0(N)\, X_1(N)$ over $\\mathbb{Z}[1/N]$ and models o
ver $\\mathbb{Z}$ à la Deligne-Rapoport and Katz-Mazur\, consequences for
the structure of the Néron model of $J_0(p)$\n
LOCATION:https://researchseminars.org/talk/STAGE/95/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hari Iyer (Harvard)
DTSTART;VALUE=DATE-TIME:20231109T210000Z
DTEND;VALUE=DATE-TIME:20231109T223000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/96
DESCRIPTION:Title: G
alois representations and modular forms\nby Hari Iyer (Harvard) as par
t of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAb
stract\nEichler-Shimura relation on the special fiber of $J_0(N)$\, associ
ating Galois representations and abelian varieties to weight 2 cusp forms.
\n
LOCATION:https://researchseminars.org/talk/STAGE/96/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kenta Suzuki (MIT)
DTSTART;VALUE=DATE-TIME:20231116T210000Z
DTEND;VALUE=DATE-TIME:20231116T223000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/97
DESCRIPTION:Title: C
riterion 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 Simons Bu
ilding.\n\nAbstract\nReduce the proof of the main theorem to showing that
there exists a rank-$0$ quotient $A$ of $J_0(p)$ such that $[0]\\ne [\\inf
ty]$ in $A$.\n\nReference: [Ma1] Section III.5\n
LOCATION:https://researchseminars.org/talk/STAGE/97/
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:20240910T020501Z
UID:STAGE/98
DESCRIPTION:Title: H
eight functions\nby Raymond van Bommel (Massachusetts Institute of Tec
hnology) 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 fields and
function fields. We will not only consider finiteness of points of bounde
d degree and height\, but also the number of such points. If time allows\,
we will consider how the heights associated to different line bundles on
a projective variety are related.\n\nThe contents of this talk are based o
n
Chapter 2 of the book. Another good source is Lang's book.\n
LOCATION:https://researchseminars.org/talk/STAGE/98/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xinyu Zhou
DTSTART;VALUE=DATE-TIME:20240222T210000Z
DTEND;VALUE=DATE-TIME:20240222T223000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/99
DESCRIPTION:Title: N
ormalized heights\nby Xinyu Zhou as part of STAGE\n\nLecture held in R
oom 2-131 in the MIT Simons Building.\n\nAbstract\nChapter 3 of Serre\, Le
ctures on the Mordell-Weil theorem\n
LOCATION:https://researchseminars.org/talk/STAGE/99/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niven T. Achenjang (MIT)
DTSTART;VALUE=DATE-TIME:20240229T210000Z
DTEND;VALUE=DATE-TIME:20240229T223000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/100
DESCRIPTION:Title:
The Mordell-Weil theorem and Chabauty's theorem\nby Niven T. Achenjang
(MIT) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Bu
ilding.\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 fi
rst part\, we will discuss the Mordell-Weil Theorem\, which states that th
e abelian group of rational points on an abelian variety $A$ defined over
a global field $K$ is finitely generated. We will show that this theorem f
ollows from some classical finiteness results in algebraic number theory a
long with the theory of heights built up in previous talks. Time permittin
g\, we will conclude the first part by proving a theorem of Neron which gi
ves an asymptotic count for the number of points of bounded height on an a
belian variety of rank $\\rho$. In the second part\, we will turn our atte
ntion towards curves of genus $g\\ge2$. For such curves $C/K$\, we will pr
ove Chabauty's Theorem that $C(K)$ is finite if $\\operatorname{rank}\\ope
ratorname{Jac}(C)(K) < g$ (finiteness of $C(K)$ is now known even when $C$
's Jacobian has large rank).\n
LOCATION:https://researchseminars.org/talk/STAGE/100/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frank Lu (Harvard)
DTSTART;VALUE=DATE-TIME:20240307T210000Z
DTEND;VALUE=DATE-TIME:20240307T223000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/101
DESCRIPTION:Title:
The Demyanenko-Manin method and Mumford's inequality\nby Frank Lu (Har
vard) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Bui
lding.\n\nAbstract\nIn this talk\, we will discuss two theorems regarding
the number of rational points on curves of genus $g \\geq 2:$ the Demyanen
ko-Manin theorem and Mumford's inequality. We will begin with the Demyanen
ko-Manin theorem\, which tells us how the existence of enough functions $f
_i: C \\rightarrow A\,$ for some abelian variety $A\,$ allows us to show t
he number of rational points on $C$ is finite. After outlining the proof o
f this theorem and discussing an application to modular curves\, we will t
hen sketch a proof of Mumford's inequality\, which gives an asymptotic bou
nd on the number of points of bounded height without knowing Falting's the
orem.\n
LOCATION:https://researchseminars.org/talk/STAGE/101/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bjorn Poonen (MIT)
DTSTART;VALUE=DATE-TIME:20240321T200000Z
DTEND;VALUE=DATE-TIME:20240321T213000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/103
DESCRIPTION:Title:
Siegel's method\nby Bjorn Poonen (MIT) as part of STAGE\n\nLecture hel
d in Room 2-131 in the MIT Simons Building.\n\nAbstract\nWe will show how
theorems about diophantine approximation (e.g.\, Roth's theorem that irrat
ional algebraic numbers cannot be approximated too well by rational number
s) can be used to prove one of the most famous theorems of 20th century ar
ithmetic geometry\, Siegel's theorem that a hyperbolic affine curve can ha
ve only finitely many integral points. The proof is ineffective\, however
: 95 years later it is still not known if there is an algorithm that takes
as input the equation of a curve and returns the list of its integral poi
nts.\n\nReference: Chapter 7 of Serre\, Lectures on the Mordell-Weil theor
em.\n
LOCATION:https://researchseminars.org/talk/STAGE/103/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shiva Chidambaram (MIT)
DTSTART;VALUE=DATE-TIME:20240404T200000Z
DTEND;VALUE=DATE-TIME:20240404T213000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/104
DESCRIPTION:Title:
Baker's method\nby Shiva Chidambaram (MIT) as part of STAGE\n\nLecture
held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nIn this talk\
, we will discuss Baker's theorem on lower bounds for linear forms in loga
rithms\, and how it gives effective bounds for quasi-integral points on $\
\mathbb{P}^1 \\setminus \\{0\,1\,\\infty\\}$. Using coverings\, this furth
er yields effective bounds for quasi-integral points on elliptic\, superel
liptic and certain hyperelliptic affine curves. We will also discuss an ap
plication towards finding elliptic curves with good reduction outside a gi
ven finite set of places.\n\nReference: Chapter 8 of Serre\, Lectures on t
he Mordell-Weil theorem.\n
LOCATION:https://researchseminars.org/talk/STAGE/104/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Hu (Harvard)
DTSTART;VALUE=DATE-TIME:20240411T200000Z
DTEND;VALUE=DATE-TIME:20240411T213000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/105
DESCRIPTION:Title:
The Hilbert irreducibility theorem\nby Daniel Hu (Harvard) as part of
STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstrac
t\nI will introduce the notion of thin sets and discuss some applications
to Galois groups of polynomials. Then\, I will state and prove Hilbert's i
rreducibility theorem following Serre's account of Lang's proof.\n\nRefere
nce: Chapter 9 of Serre\, Lectures on the Mordell-Weil theorem.\n
LOCATION:https://researchseminars.org/talk/STAGE/105/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vijay Srinivasan
DTSTART;VALUE=DATE-TIME:20240418T200000Z
DTEND;VALUE=DATE-TIME:20240418T213000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/106
DESCRIPTION:Title:
Construction of Galois extensions\nby Vijay Srinivasan as part of STAG
E\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nT
he inverse Galois problem asks whether every finite group can be realized
as the Galois group of a finite extension of $\\mathbb{Q}$. In this talk\,
we will discuss the cases of $S_n$ and $\\text{PGL}_2(\\mathbb{F}_p)$. Th
ese 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:https://researchseminars.org/talk/STAGE/106/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jane Shi
DTSTART;VALUE=DATE-TIME:20240502T200000Z
DTEND;VALUE=DATE-TIME:20240502T213000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/107
DESCRIPTION:Title:
Construction of elliptic curves of large rank\nby Jane Shi as part of
STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstrac
t\nIn this talk\, I will first prove Néron's theorem and explain how we c
an use it as a basis to construct elliptic curves of large rank. Then\, I
will discuss two methods for constructing elliptic curves of rank at 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 elliptic cu
rves of rank at least $ 11$.\n
LOCATION:https://researchseminars.org/talk/STAGE/107/
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:20240910T020501Z
UID:STAGE/108
DESCRIPTION:Title:
The large sieve\nby Daniel Larsen (Massachusetts Institute of Technolo
gy) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Build
ing.\n\nAbstract\nIn this talk\, we will prove a version of the large siev
e inequality\, a result from analytic number theory that will eventually b
e used to give bounds on thin sets. Along the way\, we will prove the Dave
nport-Halberstam theorem and generally try to understand how the support o
f a function's Fourier transform influences the function's behavior.\n\nRe
ference: Chapter 12 of Serre\, Lectures on the Mordell-Weil theorem.\nWear
ing a mask is welcomed\, but optional.\n
LOCATION:https://researchseminars.org/talk/STAGE/108/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hao Peng (MIT)
DTSTART;VALUE=DATE-TIME:20240509T200000Z
DTEND;VALUE=DATE-TIME:20240509T213000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/109
DESCRIPTION:Title:
Applications of the large sieve to thin sets\nby Hao Peng (MIT) as par
t of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAb
stract\nWe review the proof of Cohen-Serre bound on the number of rational
points on projective and affine varieties using the large sieve method an
d Lang-Weil bound on rational points on varieties over finite fields. Noti
ce that stronger bounds are known now by work of Browning\, Heath-Brown an
d Salberger.\n
LOCATION:https://researchseminars.org/talk/STAGE/109/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elia Gorokhovsky
DTSTART;VALUE=DATE-TIME:20240909T143000Z
DTEND;VALUE=DATE-TIME:20240909T160000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/110
DESCRIPTION:Title:
Complex analytic spaces\, vector bundles\, and GAGA\nby Elia Gorokhovs
ky as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Buildi
ng.\n\nAbstract\nDefinition of the category of complex analytic spaces\, a
nd statements of GAGA.\n\nReferences: \n*\n- Hartshorne\,
*Algebrai
c geometry*\, 1977. Appendix B.1 and B.2. \n- Gunning and Rossi\
,
*Analytic functions of several complex variables*\, Prentice-Hall (
1965). \n- Serre\, Gé\;ometrie algé\;brique et gé\
;ometrie analytique\,
*Ann. Inst. Fourier* **6** (1956)\, 1-42.\n - Grothendieck\, SGA I\, Exp. XII
\n

\n
LOCATION:https://researchseminars.org/talk/STAGE/110/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xinyu Fang
DTSTART;VALUE=DATE-TIME:20240916T143000Z
DTEND;VALUE=DATE-TIME:20240916T160000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/111
DESCRIPTION:Title:
Local systems\, fundamental groupoid\, definition of connection\nby Xi
nyu Fang as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons
Building.\n\nAbstract\nThe definition of local systems vs. vector bundles.
Definition of the fundamental group and fundamental groupoid of a nice t
opological space. Statement of equivalence between the category of local
systems and the category of finite-dimensional representations of the fund
amental group. Definition of connection on a vector bundle.\n\nReference:
Deligne\, up to Section 2.9.\n
LOCATION:https://researchseminars.org/talk/STAGE/111/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Atticus Wang
DTSTART;VALUE=DATE-TIME:20240923T143000Z
DTEND;VALUE=DATE-TIME:20240923T160000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/112
DESCRIPTION:Title:
Integrable connections\nby Atticus Wang as part of STAGE\n\nLecture he
ld in Room 2-449 in the MIT Simons Building.\n\nAbstract\nDefinition of cu
rvature and integrable connection. Statement of the equivalence between t
he category of local systems and the category of vector bundles with integ
rable connection. Variants: schemes\, relative setting.\n\nReference: Del
igne\, 2.10 to the end of Section 2.\n
LOCATION:https://researchseminars.org/talk/STAGE/112/
END:VEVENT
BEGIN:VEVENT
SUMMARY:TBA
DTSTART;VALUE=DATE-TIME:20240930T143000Z
DTEND;VALUE=DATE-TIME:20240930T160000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/113
DESCRIPTION:Title:
Relationship to PDEs and $n$th order differential equations\nby TBA as
part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n
\nAbstract\nReference: Deligne\, Sections 3 and 4.\n
LOCATION:https://researchseminars.org/talk/STAGE/113/
END:VEVENT
BEGIN:VEVENT
SUMMARY:TBA
DTSTART;VALUE=DATE-TIME:20241007T143000Z
DTEND;VALUE=DATE-TIME:20241007T160000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/114
DESCRIPTION:Title:
Second order differential equations and projective connections\nby TBA
as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building
.\n\nAbstract\nReference: Deligne\, Section 5.\n
LOCATION:https://researchseminars.org/talk/STAGE/114/
END:VEVENT
BEGIN:VEVENT
SUMMARY:TBA
DTSTART;VALUE=DATE-TIME:20241209T153000Z
DTEND;VALUE=DATE-TIME:20241209T170000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/115
DESCRIPTION:by TBA as part of STAGE\n\nLecture held in Room 2-449 in the M
IT Simons Building.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/STAGE/115/
END:VEVENT
BEGIN:VEVENT
SUMMARY:TBA
DTSTART;VALUE=DATE-TIME:20241021T143000Z
DTEND;VALUE=DATE-TIME:20241021T160000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/116
DESCRIPTION:by TBA as part of STAGE\n\nLecture held in Room 2-449 in the M
IT Simons Building.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/STAGE/116/
END:VEVENT
BEGIN:VEVENT
SUMMARY:TBA
DTSTART;VALUE=DATE-TIME:20241028T143000Z
DTEND;VALUE=DATE-TIME:20241028T160000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
UID:STAGE/117
DESCRIPTION:by TBA as part of STAGE\n\nLecture held in Room 2-449 in the M
IT Simons Building.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/STAGE/117/
END:VEVENT
BEGIN:VEVENT
SUMMARY:TBA
DTSTART;VALUE=DATE-TIME:20241104T153000Z
DTEND;VALUE=DATE-TIME:20241104T170000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
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:https://researchseminars.org/talk/STAGE/118/
END:VEVENT
BEGIN:VEVENT
SUMMARY:TBA
DTSTART;VALUE=DATE-TIME:20241118T153000Z
DTEND;VALUE=DATE-TIME:20241118T170000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
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:https://researchseminars.org/talk/STAGE/119/
END:VEVENT
BEGIN:VEVENT
SUMMARY:TBA
DTSTART;VALUE=DATE-TIME:20241125T153000Z
DTEND;VALUE=DATE-TIME:20241125T170000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
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:https://researchseminars.org/talk/STAGE/120/
END:VEVENT
BEGIN:VEVENT
SUMMARY:TBA
DTSTART;VALUE=DATE-TIME:20241202T153000Z
DTEND;VALUE=DATE-TIME:20241202T170000Z
DTSTAMP;VALUE=DATE-TIME:20240910T020501Z
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:https://researchseminars.org/talk/STAGE/121/
END:VEVENT
END:VCALENDAR