# Abstracts for Conference on the Cohomology of Arithmetic Groups on the occasion of Joachim Schwermer's 66th birthday

Alternatively have a look at the program.

## On the cohomology of local Shimura varieties

Posted in
Speaker:
Peter Scholze
Affiliation:
U Bonn
Date:
Mon, 2016-06-06 09:30 - 10:30
Location:
MPIM Lecture Hall

Let $G$ be a reductive group over $Q_p$. Fargues conjectured a geometric Langlands type statement
for the stack Bun_G of $G$-bundles on the Fargues-Fontaine curve, refining the conjectural local
Langlands correspondence. The conjecture requires a formalism of "constructible" $l$-adic sheaves
on Bun_G, and a geometric Satake equivalence for the B_dR-Grassmannian. We will explain some
progress towards establishing such a formalism. If time permits, we will indicate some expected
applications of the formalism to open problems in the area.

## The hyperbolic Ax-Lindemann theorem

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Speaker:
Bruno Klingler
Affiliation:
U Paris-Diderot
Date:
Mon, 2016-06-06 11:00 - 12:00
Location:
MPIM Lecture Hall

The hyperbolic Ax-Lindemann-Weierstrass conjecture is a functional algebraic independence statement
for the uniformizing map of an arithmetic variety. In this talk I will describe the conjecture, its role and
its proof (joint work with E. Ullmo and A. Yafaev).

## Vanishing for the cohomology of hyperbolic 7-manifolds coming from triality

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Speaker:
Laurent Clozel
Affiliation:
U Paris-Sud
Date:
Mon, 2016-06-06 15:00 - 16:00
Location:
MPIM Lecture Hall

This is common work with Nicolas Bergeron. The last hyperbolic compact manifolds for which
Thurston's conjecture, on the existence of covers with non-trivial $H^1$, is unknown are the manifolds
of dimension 7 associated to triality orthogonal groups. I will sketch the proof of the negative answer
when congruence groups are considered. This result was announced by us some time ago, but the proof
had to wait for the general case of the twisted trace formula, now established by Moeglin and Waldspurger.

Posted in
Speaker:
Alan Reid
Affiliation:
U of Texas, Austin
Date:
Mon, 2016-06-06 16:30 - 17:30
Location:
MPIM Lecture Hall

In his 1982 Bulletin Article, Qn 19 of the problem list, Thurston posed: "Find topological and
geometric properties of quotient spaces of arithmetic subgroups of PSL(2,C). These manifolds
often seem to have special beauty." Arithmetic link complements in the 3-sphere provide some
particularly interesting examples of this. In this talk we take up this theme, compare and contrast
with dimension 2, and report on some recent work.

## Cohomology and Reciprocity

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Speaker:
Frank Calegari
Affiliation:
U of Chicago
Date:
Tue, 2016-06-07 09:30 - 10:30
Location:
MPIM Lecture Hall

There are well known conjectures in the Langlands Programme which link regular motives to the
cohomology of arithmetic groups. The most successful approach to this problem so far comes out
of the work of Andrew Wiles. In this survey talk, I will discuss the extent to which these methods
can be generalized to situations where the underlying arithmetic group is $\mathrm{GL}_N(\mathbf{Z})$.

## On local constancy of dimension of slope subspaces of automorphic forms

Posted in
Speaker:
Joachim Mahnkopf
Affiliation:
U Wien
Date:
Tue, 2016-06-07 11:00 - 12:00
Location:
MPIM Lecture Hall

We prove a higher rank analogoue of a Conjecture of Gouvea-Mazur on local constancy of dimension of
slope subspaces of automorphic forms for reductive groups having discrete series. The proof is based on
a comparison of Bewersdorff's elementary trace formula for pairs of congruent weights and does not
make use of p-adic Banach space methods or rigid analytic geometry.

## $F_2$ cohomology of arithmetic groups applied to discrete math, computer science and topology

Posted in
Speaker:
Alex Lubotzky
Affiliation:
Hebrew University, Jerusalem/ETH-ITS Zurich
Date:
Tue, 2016-06-07 15:00 - 16:00
Location:
MPIM Lecture Hall

Expander graphs have been a focus of interest in computer science in the last 40 years and in the
last 10 years also in pure math. In recent years a high dimensional theory of expanders is emerged.
A fundamental concept  there is the coboundry expander. Cohomology of arithmetic groups
( mainly with coefficients in in $F_2$) plays an important role in these developments. We will
present some applications to topology and computer science as well as some open problems.

## A-packets for complex classical groups (joint work with David Renard)

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Speaker:
Colette Moeglin
Affiliation:
CNRS
Date:
Tue, 2016-06-07 16:30 - 17:30
Location:
MPIM Lecture Hall

I will give a full description of Arthur's packets on the field of complex numbers. In this case,
the situation is particularly simple because the classification of the unitary dual is known after
Barbasch's work.

## The "colourful gown of theory" according to Claude Chevalley

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Speaker:
Norbert Schappacher
Affiliation:
IRMA
Date:
Wed, 2016-06-08 09:30 - 10:30
Location:
MPIM Lecture Hall

The physicist Heinrich Hertz wrote in 1892 that nature - or whatever the object of our science is - never
shows herself naked, but always clad in a colouful gown whose cut and colour is completely made up
by us. Claude Chevalley's mathematical papers are dry and forbidding; they hardly show their author
as the taylor that he was. The talk will add other, rather colourful aspects to this image of the mathematician
by looking at his whole literary production from the 1930s. The plea is for a richer, and eventually more

## Plectic cohomology of Shimura varieties

Posted in
Speaker:
Anthony J. Scholl
Affiliation:
DPMMS, Cambridge
Date:
Wed, 2016-06-08 11:00 - 12:00
Location:
MPIM Lecture Hall

We will discuss some aspects of "plectic cohomology" a (largely conjectural) cohomology theory
for a certain class of Shimura varieties. This is a report on joint work with Jan Nekovář?

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