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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:20220927T054501Z
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:TBA
DTSTART;VALUE=DATE-TIME:20221108T160000Z
DTEND;VALUE=DATE-TIME:20221108T173000Z
DTSTAMP;VALUE=DATE-TIME:20220927T054501Z
UID:STAGE/68
DESCRIPTION:Title: T
orsors over finite fields\, local fields\, and global fields\nby TBA a
s 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 o
ver local fields\; local-global principle for torsors over global fields.\
n\nReference: Po
onen\, *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:20220927T054501Z
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:TBA
DTSTART;VALUE=DATE-TIME:20221122T160000Z
DTEND;VALUE=DATE-TIME:20221122T173000Z
DTSTAMP;VALUE=DATE-TIME:20220927T054501Z
UID:STAGE/70
DESCRIPTION:Title: U
nramified torsors\nby TBA as part of STAGE\n\nLecture held in Room 2-4
49 in the MIT Simons Building.\n\nAbstract\nTopic: Unramified torsors.\n\n
Reference: Poone
n\, *Rational \npoints on varieties*\, Section 6.5.7.\n
LOCATION:https://researchseminars.org/talk/STAGE/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:TBA
DTSTART;VALUE=DATE-TIME:20221129T160000Z
DTEND;VALUE=DATE-TIME:20221129T173000Z
DTSTAMP;VALUE=DATE-TIME:20220927T054501Z
UID:STAGE/71
DESCRIPTION:Title: A
n example of descent\nby TBA as part of STAGE\n\nLecture 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: Po
onen\, *Rational \npoints on varieties*\, Section 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:20220927T054501Z
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:TBA
DTSTART;VALUE=DATE-TIME:20221213T160000Z
DTEND;VALUE=DATE-TIME:20221213T173000Z
DTSTAMP;VALUE=DATE-TIME:20220927T054501Z
UID:STAGE/73
DESCRIPTION:Title: T
he étale-Brauer obstruction and insufficiency of the obstructions\nby
TBA as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Buil
ding.\n\nAbstract\nTopics: The étale-Brauer set\; comparison with the des
cent set\; insufficiency of the obstructions for a quadric bundle over a c
urve.\n\nReference: Poonen\, *Rational \npoints on varieties*\, Sections 8.5.2-8.
5.3 and 8.6.2.\n
LOCATION:https://researchseminars.org/talk/STAGE/73/
END:VEVENT
END:VCALENDAR