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BEGIN:VEVENT
SUMMARY:Nicolas Bergeron (École normale supérieure\, Paris France)
DTSTART:20201019T133000Z
DTEND:20201019T143000Z
DTSTAMP:20260404T131142Z
UID:Cohomology/1
DESCRIPTION:Title: Sczech cocycles and hyperplane arrangements 1\nby Nicolas Be
rgeron (École normale supérieure\, Paris France) as part of CRM workshop
: Arithmetic quotients of locally symmetric spaces and their cohomology\n\
nLecture held in Virtual.\n\nAbstract\nMany authors\, among which Nori\, S
czech\, Solomon\, Stevens\, or more recently Beilinson—Kings—Levin and
\nCharollois—Dasgupta—Greenberg\, have constructed different\, but rel
ated\, linear groups cocycles that are\nusually referred to as « Eisenste
in cocycles. » In these series of lectures I will explain a topological\n
construction that is a common source for all these cocycles. One interesti
ng feature of this construction is that\nstarting from a purely topologica
l class it leads to the algebraic world of meromorphic forms on hyperplane
\ncomplements in n-fold products of either the (complex) additive group\,
the multiplicative group or a (family of)\nelliptic curve(s). We will see
that eventually our construction reveals hidden relations between products
of\nelementary (rational\, trigonometric or elliptic) functions) governed
by relations between classes in the\nhomology of linear groups.\nThis is
based on a work-in-progress with Pierre Charollois\, Luis Garcia and Aksha
y Venkatesh\n
LOCATION:https://stable.researchseminars.org/talk/Cohomology/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paul Gunnells (University of Massachusetts Amherst)
DTSTART:20201019T144500Z
DTEND:20201019T154500Z
DTSTAMP:20260404T131142Z
UID:Cohomology/2
DESCRIPTION:Title: Modular symbols over function fields\nby Paul Gunnells (Univ
ersity of Massachusetts Amherst) as part of CRM workshop: Arithmetic quoti
ents of locally symmetric spaces and their cohomology\n\nLecture held in V
irtual.\n\nAbstract\nModular symbols\, due to Birch and Manin\, provide a
very\nconcrete way to compute with classical holomorphic modular forms.\nL
ater modular symbols were extended to GL(n) by Ash and Rudolph\, and\nsinc
e then such symbols and variations have played a central role in\ncomputat
ional investigation of the cohomology of arithmetic groups\nover number fi
elds\, and in particular in explicitly computing the\nHecke action on coho
mology. \n\nA theory of modular symbols for GL(2) over the rational functi
on field\nwas developed by Teitelbaum and later applied by Armana. In thi
s talk\nwe extend this construction to GL(n) and show how it can be used t
o\ncompute Hecke operators on cohomology. This is joint work with Dan\nYa
saki.\n
LOCATION:https://stable.researchseminars.org/talk/Cohomology/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Romyar Sharifi (UCLA)
DTSTART:20201019T160000Z
DTEND:20201019T170000Z
DTSTAMP:20260404T131142Z
UID:Cohomology/3
DESCRIPTION:Title: Eisenstein cocycles in motivic cohomology\nby Romyar Sharifi
(UCLA) as part of CRM workshop: Arithmetic quotients of locally symmetric
spaces and their cohomology\n\nLecture held in Virtual.\n\nAbstract\nI wi
ll describe joint work with Akshay Venkatesh on the construction and study
of GL_2(Z)-cocycles valued in\nsecond K-groups of the function fields of
squares of the multiplicative group over the rationals and of a\nuniversal
elliptic curve over a modular curve. I will also describe the pullbacks o
f their restrictions to\ncongruence subgroups under torsion sections\, rel
ating these specializations to explicit maps on homology\ngroups of modula
r curves\n
LOCATION:https://stable.researchseminars.org/talk/Cohomology/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolas Bergeron (École normale supérieure\, Paris France)
DTSTART:20201020T133000Z
DTEND:20201020T143000Z
DTSTAMP:20260404T131142Z
UID:Cohomology/4
DESCRIPTION:Title: Sczech cocycles and hyperplane arrangements 2\nby Nicolas Be
rgeron (École normale supérieure\, Paris France) as part of CRM workshop
: Arithmetic quotients of locally symmetric spaces and their cohomology\n\
nLecture held in Virtual.\n\nAbstract\nMany authors\, among which Nori\, S
czech\, Solomon\, Stevens\, or more recently Beilinson—Kings—Levin and
\nCharollois—Dasgupta—Greenberg\, have constructed different\, but rel
ated\, linear groups cocycles that are\nusually referred to as « Eisenste
in cocycles. » In these series of lectures I will explain a topological\n
construction that is a common source for all these cocycles. One interesti
ng feature of this construction is that\nstarting from a purely topologica
l class it leads to the algebraic world of meromorphic forms on hyperplane
\ncomplements in n-fold products of either the (complex) additive group\,
the multiplicative group or a (family of)\nelliptic curve(s). We will see
that eventually our construction reveals hidden relations between products
of\nelementary (rational\, trigonometric or elliptic) functions) governed
by relations between classes in the\nhomology of linear groups.\nThis is
based on a work-in-progress with Pierre Charollois\, Luis Garcia and Aksha
y Venkatesh\n
LOCATION:https://stable.researchseminars.org/talk/Cohomology/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emmanuel Lecouturier (Yau Mathematical Sciences Center & IAS)
DTSTART:20201020T150000Z
DTEND:20201020T160000Z
DTSTAMP:20260404T131142Z
UID:Cohomology/5
DESCRIPTION:Title: On Sharifi's conjecture and generalizations\nby Emmanuel Lec
outurier (Yau Mathematical Sciences Center & IAS) as part of CRM workshop:
Arithmetic quotients of locally symmetric spaces and their cohomology\n\n
Lecture held in Virtual.\n\nAbstract\nRomyar Sharifi made beautiful explic
it conjectures relating the K-theory of cyclotomic field to modular\nsymbo
ls modulo some Eisenstein ideal.\nWe report on some partial results on the
se conjectures and their implication for Mazur's Eisenstein ideal. We\nals
o discuss some ongoing project exploring the analogue of these conjectures
for Bianchi manifolds. All this\nis joint work with Jun Wang\n
LOCATION:https://stable.researchseminars.org/talk/Cohomology/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolas Bergeron (École normale supérieure\, Paris France)
DTSTART:20201021T133000Z
DTEND:20201021T143000Z
DTSTAMP:20260404T131142Z
UID:Cohomology/6
DESCRIPTION:Title: Sczech cocycles and hyperplane arrangements 3\nby Nicolas Be
rgeron (École normale supérieure\, Paris France) as part of CRM workshop
: Arithmetic quotients of locally symmetric spaces and their cohomology\n\
nLecture held in Virtual.\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/Cohomology/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Johannes Sprang (Universität Regensburg)
DTSTART:20201021T144500Z
DTEND:20201021T154500Z
DTSTAMP:20260404T131142Z
UID:Cohomology/7
DESCRIPTION:Title: The Equivariant Polylogarithm and Eisenstein classes\nby Joh
annes Sprang (Universität Regensburg) as part of CRM workshop: Arithmetic
quotients of locally symmetric spaces and their cohomology\n\nLecture hel
d in Virtual.\n\nAbstract\nIn this lecture\, I will report on recent resul
ts\, joint with Guido Kings\, on the construction of equivariant\nEisenste
in classes. The equivariant polylogarithm is a very general tool for const
ructing motivic cohomology\nclasses of arithmetic groups. A certain refine
ment of the de Rham realization of these classes gives interesting\nalgebr
aic Eisenstein classes. As an application of our construction\, we prove a
lgebraicity results for critical\nHecke L-values of totally imaginary fiel
ds. This generalizes previous results of Damerell\, Shimura and Katz in\nt
he CM case. The integrality of our construction allows us to construct p-a
dic L-functions for totally imaginary\nfields at ordinary primes.\n
LOCATION:https://stable.researchseminars.org/talk/Cohomology/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Garcia Martinez (University College London)
DTSTART:20201021T160000Z
DTEND:20201021T170000Z
DTSTAMP:20260404T131142Z
UID:Cohomology/8
DESCRIPTION:Title: An Eisenstein cocycle for imaginary quadratic fields\nby Lui
s Garcia Martinez (University College London) as part of CRM workshop: Ari
thmetic quotients of locally symmetric spaces and their cohomology\n\nLect
ure held in Virtual.\n\nAbstract\nI will give details of the general pictu
re discussed by Nicolas Bergeron in the case of arithmetic subgroups of\nS
L_N(k)\, where k is an imaginary quadratic field. I will introduce a cocyc
le for such groups whose values are\npolynomials in classical Kronecker—
Eisenstein series. We will then see how this cocycle leads to explicit\nfo
rmulas for critical values of Hecke L—functions of degree N extensions o
f k\, generalising work of Colmez.\nThis is based on a work-in-progress wi
th Pierre Charollois\, Luis Garcia and Akshay Venkatesh.\n
LOCATION:https://stable.researchseminars.org/talk/Cohomology/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolas Bergeron (École normale supérieure\, Paris France)
DTSTART:20201022T133000Z
DTEND:20201022T143000Z
DTSTAMP:20260404T131142Z
UID:Cohomology/9
DESCRIPTION:Title: Sczech cocycles and hyperplane arrangements 4\nby Nicolas Be
rgeron (École normale supérieure\, Paris France) as part of CRM workshop
: Arithmetic quotients of locally symmetric spaces and their cohomology\n\
nLecture held in Virtual.\n\nAbstract\nMany authors\, among which Nori\, S
czech\, Solomon\, Stevens\, or more recently Beilinson—Kings—Levin and
\nCharollois—Dasgupta—Greenberg\, have constructed different\, but rel
ated\, linear groups cocycles that are\nusually referred to as « Eisenste
in cocycles. » In these series of lectures I will explain a topological\n
construction that is a common source for all these cocycles. One interesti
ng feature of this construction is that\nstarting from a purely topologica
l class it leads to the algebraic world of meromorphic forms on hyperplane
\ncomplements in n-fold products of either the (complex) additive group\,
the multiplicative group or a (family of)\nelliptic curve(s). We will see
that eventually our construction reveals hidden relations between products
of\nelementary (rational\, trigonometric or elliptic) functions) governed
by relations between classes in the\nhomology of linear groups.\nThis is
based on a work-in-progress with Pierre Charollois\, Luis Garcia and Aksha
y Venkatesh.\n
LOCATION:https://stable.researchseminars.org/talk/Cohomology/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Preston Wake (Michigan State University)
DTSTART:20201022T150000Z
DTEND:20201022T160000Z
DTSTAMP:20260404T131142Z
UID:Cohomology/10
DESCRIPTION:Title: Torsion in the cohomology of Hilbert modular surfaces\nby P
reston Wake (Michigan State University) as part of CRM workshop: Arithmeti
c quotients of locally symmetric spaces and their cohomology\n\nLecture he
ld in Virtual.\n\nAbstract\nWe investigate the analogue of Mazur's Eisenst
ein ideal for Hilbert modular forms over a real quadratic field.\nUnlike i
n the case of modular forms\, we show that\, even in weight two\, there ar
e mod-p modular forms that\ndon't lift to characteristic zero. We explain
this by computing the torsion in the cohomology of the Hilbert\nmodular su
rface. This is joint work with Akshay Venkatesh.\n
LOCATION:https://stable.researchseminars.org/talk/Cohomology/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolas Bergeron (École normale supérieure\, Paris France)
DTSTART:20201023T133000Z
DTEND:20201023T143000Z
DTSTAMP:20260404T131142Z
UID:Cohomology/11
DESCRIPTION:Title: Sczech cocycles and hyperplane arrangements 5\nby Nicolas B
ergeron (École normale supérieure\, Paris France) as part of CRM worksho
p: Arithmetic quotients of locally symmetric spaces and their cohomology\n
\nLecture held in Virtual.\n\nAbstract\nMany authors\, among which Nori\,
Sczech\, Solomon\, Stevens\, or more recently Beilinson—Kings—Levin an
d\nCharollois—Dasgupta—Greenberg\, have constructed different\, but re
lated\, linear groups cocycles that are\nusually referred to as « Eisenst
ein cocycles. » In these series of lectures I will explain a topological\
nconstruction that is a common source for all these cocycles. One interest
ing feature of this construction is that\nstarting from a purely topologic
al class it leads to the algebraic world of meromorphic forms on hyperplan
e\ncomplements in n-fold products of either the (complex) additive group\,
the multiplicative group or a (family of)\nelliptic curve(s). We will see
that eventually our construction reveals hidden relations between product
s of\nelementary (rational\, trigonometric or elliptic) functions) governe
d by relations between classes in the\nhomology of linear groups.\nThis is
based on a work-in-progress with Pierre Charollois\, Luis Garcia and Aksh
ay Venkatesh.\n
LOCATION:https://stable.researchseminars.org/talk/Cohomology/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pierre Charollois (Université Paris 6)
DTSTART:20201023T150000Z
DTEND:20201023T160000Z
DTSTAMP:20260404T131142Z
UID:Cohomology/12
DESCRIPTION:Title: Trigonometric and elliptic cocycles overGLN(Z)\nby Pierre C
harollois (Université Paris 6) as part of CRM workshop: Arithmetic quotie
nts of locally symmetric spaces and their cohomology\n\nLecture held in Vi
rtual.\n\nAbstract\nI will first recall a joint work with Dasgupta and Gre
enberg (2016)\, where we elaborate on Shintani's method\nto construct an E
isenstein cocycle over GLN(<Z) taking values in a
ring of\nrational generating series that can be expressed in terms of bas
ic trigonometric functions. We establish that it\nis cohomologous to a for
mer cocycle of Sczech. After smoothing it enjoys nice integral properties.
Combined\nwith evaluation on a tori\, it allows us to recover the basic p
roperties of the Cassou-Noguès p-adic zeta\nfunctions attached to totally
real fields.\nThen I'll give an overview of the elliptic generalization I
have later obtained\, where trigonometric functions are\nnow replaced by
the Kronecker-Eisenstein function\, which is a generating series for modul
ar forms. The action\nof Hecke operators over GLN on tha
t new cocycle has been studied by Hao Zhang in\nhis thesis (2020)\, and th
e end of talk present some of his results too.\n
LOCATION:https://stable.researchseminars.org/talk/Cohomology/12/
END:VEVENT
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