# 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

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

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

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.

## Arithmetic link complements

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

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

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

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)

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

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

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?