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SUMMARY:Ken Ribet (UC Berkeley)
DTSTART;VALUE=DATE-TIME:20210105T000000Z
DTEND;VALUE=DATE-TIME:20210105T005000Z
DTSTAMP;VALUE=DATE-TIME:20240329T094240Z
UID:CHAT/1
DESCRIPTION:Title: Lan
glands correspondence and geometry\nby Ken Ribet (UC Berkeley) as part
of CHAT (Career\, History And Thoughts) series\n\n\nAbstract\nThe Langlan
ds program suggests innumerable problems in\narithmetic geometry. One cla
ss of problems concerns geometric and\ncohomological relations between alg
ebraic varieties in case such\nrelations are predicted by a Langlands corr
espondence between spaces\nof automorphic forms for different algebraic gr
oups. 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 variety in
one characteristic\nand the behavior of a second Shimura variety in a seco
nd\ncharacteristic.\n
LOCATION:https://researchseminars.org/talk/CHAT/1/
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SUMMARY:Benedict Gross (UCSD)
DTSTART;VALUE=DATE-TIME:20210112T000000Z
DTEND;VALUE=DATE-TIME:20210112T005000Z
DTSTAMP;VALUE=DATE-TIME:20240329T094240Z
UID:CHAT/2
DESCRIPTION:Title: The
conjectures of Gan\, Gross\, and Prasad\nby Benedict Gross (UCSD) as
part of CHAT (Career\, History And Thoughts) series\n\n\nAbstract\nI will
review the conjectures I made with Wee Teck Gan and Dipendra Prasad\, whic
h provide a bridge between number theory and representation theory. Beside
s stating the various conjectures and reviewing the main results that have
been obtained in this direction\, I'll make some historical remarks on ho
w we came to formulate them.\n
LOCATION:https://researchseminars.org/talk/CHAT/2/
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SUMMARY:Michael Harris (Columbia)
DTSTART;VALUE=DATE-TIME:20210202T000000Z
DTEND;VALUE=DATE-TIME:20210202T005000Z
DTSTAMP;VALUE=DATE-TIME:20240329T094240Z
UID:CHAT/3
DESCRIPTION:Title: Gal
ois representations and torsion cohomology: a series of misunderstandings<
/a>\nby Michael Harris (Columbia) as part of CHAT (Career\, History And Th
oughts) series\n\n\nAbstract\nIn 2013\, Peter Scholze announced his proof
that Galois representations with finite coefficients could be associated t
o \ntorsion classes in the cohomology of certain locally symmetric spaces.
The existence of such a\ncorrespondence had been predicted by a number o
f 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 starts and
occasional successes\, and on my own intermittent involvement\nwith this a
nd related problems.\n
LOCATION:https://researchseminars.org/talk/CHAT/3/
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SUMMARY:Barry Mazur (Harvard)
DTSTART;VALUE=DATE-TIME:20210405T230000Z
DTEND;VALUE=DATE-TIME:20210405T235000Z
DTSTAMP;VALUE=DATE-TIME:20240329T094240Z
UID:CHAT/4
DESCRIPTION:Title: Tho
ughts about Primes and Knots\nby Barry Mazur (Harvard) as part of CHAT
(Career\, History And Thoughts) series\n\n\nAbstract\nKnots and their exq
uisitely idiosyncratic properties\, are the vital essence of three-dimensi
onal\ntopology. Primes and their exquisitely idiosyncratic properties\, ar
e the vital essence of number\ntheory. A striking (and extremely useful) a
nalogy between Knots and Primes helped me as I\nbecame as passionate about
number theory as I was (and still am) about knots. I’m delighted\nto ha
ve been asked by Shekhar and Chi-Yun to be part of the ‘experiment’ in
this (experimental)\nseries of talks: CHAT: Career\, History and Thoughts
to think again about this\, and take part\nin a Q&A with people in the se
minar.\n
LOCATION:https://researchseminars.org/talk/CHAT/4/
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SUMMARY:Peter Sarnak (Princeton)
DTSTART;VALUE=DATE-TIME:20210517T230000Z
DTEND;VALUE=DATE-TIME:20210517T235000Z
DTSTAMP;VALUE=DATE-TIME:20240329T094240Z
UID:CHAT/5
DESCRIPTION:Title: Aut
omorphic Cuspidal Representations and Maass Forms\nby Peter Sarnak (Pr
inceton) as part of CHAT (Career\, History And Thoughts) series\n\n\nAbstr
act\nThe building blocks for automorphic representations on $\\mathrm{GL}_
n$ are the cusp forms. Even the existence of Maass cusp forms is subtle an
d tied to arithmetic. I will describe some of my many encounters with thes
e trascendental objects and speculate about their role in number theory.\n
LOCATION:https://researchseminars.org/talk/CHAT/5/
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BEGIN:VEVENT
SUMMARY:Henri Darmon (McGill)
DTSTART;VALUE=DATE-TIME:20210524T230000Z
DTEND;VALUE=DATE-TIME:20210524T235000Z
DTSTAMP;VALUE=DATE-TIME:20240329T094240Z
UID:CHAT/6
DESCRIPTION:Title: Mod
ular functions and explicit class field theory: private reminiscences and
public confessions\nby Henri Darmon (McGill) as part of CHAT (Career\,
History And Thoughts) series\n\n\nAbstract\nThe problem of constructing c
lass fields of number fields from explicit values of modular functions ha
s its roots in the theory of cyclotomic fields and the theory of complex m
ultiplication. The latter theory acquired a renewed currency in the second
half of the 20th century through its connections to the arithmetic of ell
iptic curves\, manifested in the work of Coates--Wiles\, Rubin\, Gross--Za
gier\, and Kolyvagin. \n\nI will give a personal account of my path towar
ds a (slightly) better understanding of explicit class field theory for re
al quadratic fields and its applications to elliptic curves\, taking advan
tage of the CHAT format to focus on the misconceptions\, false starts\, an
d dead ends that have marked my roundabout and tortuous\, but also very en
joyable\, mathematical journey so far.\n
LOCATION:https://researchseminars.org/talk/CHAT/6/
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SUMMARY:Hélène Esnault (FU Berlin/Harvard/Copenhagen)
DTSTART;VALUE=DATE-TIME:20231204T230000Z
DTEND;VALUE=DATE-TIME:20231204T235000Z
DTSTAMP;VALUE=DATE-TIME:20240329T094240Z
UID:CHAT/7
DESCRIPTION:Title: Cod
imension one in Algebraic and Arithmetic Geometry\nby Hélène Esnault
(FU Berlin/Harvard/Copenhagen) as part of CHAT (Career\, History And Thou
ghts) series\n\n\nAbstract\nThe notions of $\\textit{weight}$ in complex g
eometry and in $\\ell$-adic theory in geometry over a finite field\nhave b
een developed by Deligne and by the Grothendieck school. The analogy betw
een the theories is foundational \nand led to predictions and theorems o
n both sides. \nOn the complex Hodge theory side\, not only do we have 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 documen
ted by Deligne. \nThinking of this gave the way to understand the Lang--Ma
nin conjecture according to which smooth projective\n rationally connected
varieties over a finite field possess a rational point. \n$\\url{http://p
age.mi.fu-berlin.de/esnault/preprints/helene/62-chowgroup.pdf}$\n\nOn the
other hand\, we know the formulation in complex geometry of the Hodge conj
ecture: on a smooth projective complex variety $X$\, \na sub-Hodge structu
re of $H^{2j}(X)$ of Hodge type $(j\,j)$ should be supported on a codi
mension $j$ cycle. The analog $\\ell$-adic conjecture \nhas been formulat
ed 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 die at t
he generic point of the variety. \nThis is difficult to formulate because
Hodge structures are complicated to describe. But there is one instance fo
r 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 conjecture de
scends to the field of definition of $X$ and becomes purely algebraic. \nI
t is on the one hand related to the (quite bold) motivic conjectures predi
cting that $H^i(X\,\\mathcal O)=0$ for all $i\\neq 0$ should be equivalent
to the triviality of the Chow group of $0$-cycles over a large field (thi
s brings us back to the proof of the Lang--Manin conjecture). On the other
hand\, as it is purely algebraic\, one can try to think of it in the fram
ework of today’s $p$-adic Hodge theory.\n
LOCATION:https://researchseminars.org/talk/CHAT/7/
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