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SUMMARY:Ken Ribet (UC Berkeley)
DTSTART:20210105T000000Z
DTEND:20210105T005000Z
DTSTAMP:20260404T094534Z
UID:CHAT/1
DESCRIPTION:Title: Langlands correspondence and geometry\nby Ken Ribet (UC Berkeley)
as part of CHAT (Career\, History And Thoughts) series\n\n\nAbstract\nThe
Langlands program suggests innumerable problems in\narithmetic geometry.
One class of problems concerns geometric and\ncohomological relations betw
een algebraic varieties in case such\nrelations are predicted by a Langlan
ds correspondence between spaces\nof automorphic forms for different algeb
raic groups. I will describe\nhow my quest for such a relation led me to
realize that there can be a\nlink between the behavior of one Shimura vari
ety in one characteristic\nand the behavior of a second Shimura variety in
a second\ncharacteristic.\n
LOCATION:https://stable.researchseminars.org/talk/CHAT/1/
END:VEVENT
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SUMMARY:Benedict Gross (UCSD)
DTSTART:20210112T000000Z
DTEND:20210112T005000Z
DTSTAMP:20260404T094534Z
UID:CHAT/2
DESCRIPTION:Title: The conjectures of Gan\, Gross\, and Prasad\nby Benedict Gross (UC
SD) as part of CHAT (Career\, History And Thoughts) series\n\n\nAbstract\n
I will review the conjectures I made with Wee Teck Gan and Dipendra Prasad
\, which provide a bridge between number theory and representation theory.
Besides stating the various conjectures and reviewing the main results th
at have been obtained in this direction\, I'll make some historical remark
s on how we came to formulate them.\n
LOCATION:https://stable.researchseminars.org/talk/CHAT/2/
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SUMMARY:Michael Harris (Columbia)
DTSTART:20210202T000000Z
DTEND:20210202T005000Z
DTSTAMP:20260404T094534Z
UID:CHAT/3
DESCRIPTION:Title: Galois representations and torsion cohomology: a series of misundersta
ndings\nby Michael Harris (Columbia) as part of CHAT (Career\, History
And Thoughts) series\n\n\nAbstract\nIn 2013\, Peter Scholze announced his
proof that Galois representations with finite coefficients could be assoc
iated to \ntorsion classes in the cohomology of certain locally symmetric
spaces. The existence of such a\ncorrespondence had been predicted by a n
umber of mathematicians but for a long time no one had the slightest idea\
nhow to construct the Galois representations. In this talk I will review
some of the history of the problem\, with\nemphasis on the many false star
ts and occasional successes\, and on my own intermittent involvement\nwith
this and related problems.\n
LOCATION:https://stable.researchseminars.org/talk/CHAT/3/
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SUMMARY:Barry Mazur (Harvard)
DTSTART:20210405T230000Z
DTEND:20210405T235000Z
DTSTAMP:20260404T094534Z
UID:CHAT/4
DESCRIPTION:Title: Thoughts about Primes and Knots\nby Barry Mazur (Harvard) as part
of CHAT (Career\, History And Thoughts) series\n\n\nAbstract\nKnots and th
eir exquisitely idiosyncratic properties\, are the vital essence of three-
dimensional\ntopology. Primes and their exquisitely idiosyncratic properti
es\, are the vital essence of number\ntheory. A striking (and extremely us
eful) analogy between Knots and Primes helped me as I\nbecame as passionat
e about number theory as I was (and still am) about knots. I’m delighted
\nto have been asked by Shekhar and Chi-Yun to be part of the ‘experimen
t’ in this (experimental)\nseries of talks: CHAT: Career\, History and T
houghts to think again about this\, and take part\nin a Q&A with people in
the seminar.\n
LOCATION:https://stable.researchseminars.org/talk/CHAT/4/
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SUMMARY:Peter Sarnak (Princeton)
DTSTART:20210517T230000Z
DTEND:20210517T235000Z
DTSTAMP:20260404T094534Z
UID:CHAT/5
DESCRIPTION:Title: Automorphic Cuspidal Representations and Maass Forms\nby Peter Sar
nak (Princeton) as part of CHAT (Career\, History And Thoughts) series\n\n
\nAbstract\nThe building blocks for automorphic representations on $\\math
rm{GL}_n$ are the cusp forms. Even the existence of Maass cusp forms is su
btle and tied to arithmetic. I will describe some of my many encounters wi
th these trascendental objects and speculate about their role in number th
eory.\n
LOCATION:https://stable.researchseminars.org/talk/CHAT/5/
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BEGIN:VEVENT
SUMMARY:Henri Darmon (McGill)
DTSTART:20210524T230000Z
DTEND:20210524T235000Z
DTSTAMP:20260404T094534Z
UID:CHAT/6
DESCRIPTION:Title: Modular functions and explicit class field theory: private reminiscenc
es and public confessions\nby Henri Darmon (McGill) as part of CHAT (C
areer\, History And Thoughts) series\n\n\nAbstract\nThe problem of constru
cting class fields of number fields from explicit values of modular funct
ions has its roots in the theory of cyclotomic fields and the theory of co
mplex multiplication. The latter theory acquired a renewed currency in the
second half of the 20th century through its connections to the arithmetic
of elliptic curves\, manifested in the work of Coates--Wiles\, Rubin\, Gr
oss--Zagier\, and Kolyvagin. \n\nI will give a personal account of my pat
h towards a (slightly) better understanding of explicit class field theory
for real quadratic fields and its applications to elliptic curves\, takin
g advantage of the CHAT format to focus on the misconceptions\, false star
ts\, and dead ends that have marked my roundabout and tortuous\, but also
very enjoyable\, mathematical journey so far.\n
LOCATION:https://stable.researchseminars.org/talk/CHAT/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hélène Esnault (FU Berlin/Harvard/Copenhagen)
DTSTART:20231204T230000Z
DTEND:20231204T235000Z
DTSTAMP:20260404T094534Z
UID:CHAT/7
DESCRIPTION:Title: Codimension one in Algebraic and Arithmetic Geometry\nby Hélène
Esnault (FU Berlin/Harvard/Copenhagen) as part of CHAT (Career\, History A
nd Thoughts) series\n\n\nAbstract\nThe notions of $\\textit{weight}$ in co
mplex geometry and in $\\ell$-adic theory in geometry over a finite field\
nhave been developed by Deligne and by the Grothendieck school. The analo
gy between the theories is foundational \nand led to predictions and the
orems on both sides. \nOn the complex Hodge theory side\, not only do we h
ave the weight filtration\, but we also have the Hodge filtration. \nThe
analogy on the $\\ell$-adic side over a finite field hasn’t really been
documented by Deligne. \nThinking of this gave the way to understand the L
ang--Manin conjecture according to which smooth projective\n rationally co
nnected varieties over a finite field possess a rational point. \n$\\url{h
ttp://page.mi.fu-berlin.de/esnault/preprints/helene/62-chowgroup.pdf}$\n\n
On the other hand\, we know the formulation in complex geometry of the Hod
ge conjecture: on a smooth projective complex variety $X$\, \na sub-Hodge
structure of $H^{2j}(X)$ of Hodge type $(j\,j)$ should be supported on
a codimension $j$ cycle. The analog $\\ell$-adic conjecture \nhas been f
ormulated by Tate\, even over a number field. Grothendieck’s generalized
Hodge conjecture is straightforwardly formulated: \na sub-Hodge structure
$H$ of $H^i(X)$ of Hodge type $(i-1\,1)\, (i-2\,2)\, \\ldots\, (1\,i-1)$
should be supported on a codimension $1$ cycle. \nEquivalently it should d
ie at the generic point of the variety. \nThis is difficult to formulate b
ecause Hodge structures are complicated to describe. But there is one inst
ance for which we can bypass the Hodge formulation:\n$H=H^i(X)$ and $H^{0\
,i}=H^i(X\, \\mathcal O)(=H^{i\,0}=H^0(X\, \\Omega^i))=0$. Then the conjec
ture descends to the field of definition of $X$ and becomes purely algebra
ic. \nIt is on the one hand related to the (quite bold) motivic conjecture
s predicting that $H^i(X\,\\mathcal O)=0$ for all $i\\neq 0$ should be equ
ivalent to the triviality of the Chow group of $0$-cycles over a large fie
ld (this brings us back to the proof of the Lang--Manin conjecture). On th
e other hand\, as it is purely algebraic\, one can try to think of it in t
he framework of today’s $p$-adic Hodge theory.\n
LOCATION:https://stable.researchseminars.org/talk/CHAT/7/
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