Vorträge des SFB 1442

Pro-etale cohomology of rigid-analytic spaces with Q_p-coefficients has some surprising features: it is not A^1-invariant and no general finiteness theorems over Q_p are true. It has been observed in recent years that these particularities can be explained by viewing the pro-etale cohomology as (quasi-)coherent cohomology on the Fargues-Fontaine curve. I want to explain joint work in progress with Arthur-Cesar Le Bras and Lucas Mann, which aims to fully implement this idea by developing a six functor formalism with values in solid quasi-coherent sheaves on relative Fargues-Fontaine curves.

Already Euler computed the values $\zeta(2), \zeta(4), \zeta(6),\ldots$ of the Riemann zeta function $\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$ to be \begin{equation*} \zeta(2k)=-\frac{(2\pi i)^{2k}}{2(2k!)}B_{2k} \end{equation*} where $B_{2k}\in \mathbb{Q}$ are the Bernoulli numbers. This formula can be seen as the easiest case of a vast conjecture by Deligne from 1977, which relates special values of $L$-functions of arithmetic varieties and their periods. In this talk we want to give a non-technical introduction to the Deligne conjecture, aimed at general mathematical audience. In the end we discuss very recent developments, which lead to a complete proof in the case of Hecke $L$-functions.

Usual (=Euclidean) billiard is physically motivated by a variational principle based on the length of cords. Replacing length by (symplectic) area leads to symplectic billiard. Through examples and pictures we will discuss first properties of and results for symplectic billiards for smooth curves as well as for polygons. Symplectic billiard has also a curious link to basic geometric approximation theory. Then we will see polygons on which symplectic billiards have surprising dynamical properties none of which are possible for Euclidean billiards. In the end I will present a theorem giving sufficient criteria for polygons to exhibit these dynamical properties. This is joint work with Sergei Tabachnikov, and Fabian Lander and Jannik Westermann.