## p-adic limits of renormalized logarithmic Euler characteristics

Given a countable residually finite group $\Gamma$, we write $\Gamma_n \to e$ if $(\Gamma_n)$ is a sequence of normal subgroups of finite index such that any infinite intersection of $\Gamma_n$'s contains only the unit element $e$ of $\Gamma$. Given a $\Gamma$-module $M$ we are interested in the multiplicative Euler characteristics \begin{equation} \chi (\Gamma_n , M) = \prod_i |H_i(\Gamma_n , M)|^{(-1)^i} \end{equation} and the limit in the field $\mathbb{Q}_p$ of $p$-adic numbers \begin{equation} h_p := \lim_{n\to\infty} (\Gamma : \Gamma_n)^{-1} \log_p \chi (\Gamma_n , M) \; .

## Rational points on Picard modular surfaces

Picard modular surfaces X, which are smooth compactifications of the quotients Y of the complex ball by a discrete subgroups \Gamma of SU(2,1), have been studied from various points of view. They are often defined over an imaginary quadratic field M, and we are interested in the rational points of X over finite extensions k of M. In a joint work with M.

## Tilting modules for reductive groups and the Hecke category, III

Determining the characters of indecomposable tilting modules for reductive groups is one of the most fundamental

open problems in modular representation theory. A solution for GL_n would answer the question of the dimensions

of the simple modules for the symmetric group in characteristic p.

It is related to (but almost certainly harder than) the determination of the simple characters. I will describe a new

algorithm (based on a long and ongoing series of work with Elias, Riche, Libedinsky, Achar-Makisumi-Riche)

## Tilting modules for reductive groups and the Hecke category, II

Determining the characters of indecomposable tilting modules for reductive groups is one of the most fundamental

open problems in modular representation theory. A solution for GL_n would answer the question of the dimensions

of the simple modules for the symmetric group in characteristic p.

It is related to (but almost certainly harder than) the determination of the simple characters. I will describe a new

algorithm (based on a long and ongoing series of work with Elias, Riche, Libedinsky, Achar-Makisumi-Riche)

## Cabling and the solvable filtration on knot concordance

## New guests at the MPIM

## Introduction to the Thurston norm

## Topological field theory on r-spin surfaces and the Arf invariant

We present a state-sum construction of topological field theories (TFT)

on r-spin surfaces which uses a combinatorial model of r-spin structures

based on the work of Novak. We give an example of such a TFT which

computes the Arf invariant for r even. We use the combinatorial model

and this TFT to calculate diffeomorphism classes of r-spin surfaces with

parametrized boundary, extending results of Geiges and Gonzalo and of

Randal-Williams.

## Prime and squarefree values of polynomials in moderately many variables

We will present a generalisation of Schinzel's hypothesis and of the Bateman-Horn's conjecture concerning

prime values of a system of polynomials in one variable to the case of a integer form in many variables.

In particular, we will establish in this talk that a polynomial in moderately many variables takes infinitely

many prime (but also squarefree) values under some necessary assumptions. The proof will rely on Birch's

circle method and will be achieved in 50% fewer variables than in the classical Birch setting. Moreover it

## tba

## Dwork crystals and related congruences

In the talk I will describe a realization of the p-adic cohomology of an affine toric hypersurface which originates in

Dwork's work and give an explicit description of the unit-root subcrystal based on certain congruences for the coeficient

of powers of a Laurent polynomial. This is joint work with Frits Beukers. The formulas for the Frobenius and connection

on the unit-root subcrystal that we give were conjectured in my preprint ``Higher Hasse--Witt matrices'' (arXiv:1605.06440).

## Backström algebras, Backström curves and their Auslander envelopes

## From Projective Resolutions to Stable Derivators

## Exotic smooth structures on R^4, II

## Bridging the algebraic and analytic theory of the representation problem for quadratic polynomials

Given a polynomial f(x) of several variables with rational coefficients and an integer n, we say that f represents n if the equation f(x)=n is solvable in the integers. One might ask, is it possible to effectively determine the set of integers represented by f? This so-called representation problem for quadratic polynomials is one of the classical problems in number theory. The negative answer to Hilbert's 10th problem tells us that in general, there is no finite algorithm to decide whether a solution exists.

## Tilting modules for reductive groups and the Hecke category, I

Determining the characters of indecomposable tilting modules for reductive groups is one of the most fundamental

open problems in modular representation theory. A solution for GL_n would answer the question of the dimensions

of the simple modules for the symmetric group in characteristic p.

It is related to (but almost certainly harder than) the determination of the simple characters. I will describe a new

algorithm (based on a long and ongoing series of work with Elias, Riche, Libedinsky, Achar-Makisumi-Riche)

## Integral points on generalised affine Châtelet surfaces

Building up on the work of Colliot-Thélène and Sansuc which

suggests the use of Schinzel's hypothesis we show that the integral

Brauer-Manin obstruction is the only obstruction to the integral Hasse

principle for an infinite family of generalised affine Châtelet surfaces.

Moreover, we show that the set of integral points on any surface in this

family for which there is no integral Brauer-Manin obstruction satisfies a

strong approximation property away from infinity. We do so by exploiting

the conic bundle structure of such surfaces over the affine line. A