Vorträge des SFB 1442

Abstract: The space of all smooth hypersurfaces of a given degree in a complex projective variety is itself a variety. I will explain how to understand some of its cohomology using tools from homotopy theory: smooth hypersurfaces are given by holomorphic sections of line bundles subject to a differential condition, and these can be compared to continuous sections of appropriate jet bundles. This strategy usually goes under the name of "h-principle", and we will see an algebraic instance of that. I will also explain some consequences: a cohomological stability phenomenon, and a relation to classifying spaces of diffeomorphism groups of hypersurfaces.

Central sequences play a fundamental role in operator algebras, from property Gamma and the MacDuff property for II_1 factors to characterizations of tensorial absorption for C*-algebras, for example. Given the deep link between operator algebras and dynamics, it is natural to look for dynamical analogues of central sequence properties. This turns out to be subtle. In joint work with Grigorios Kopsacheilis, Hung-Chang Liao, and Andrea Vaccaro, a central sequence property (uniform property Gamma) is shown to be equivalent to a topological dynamical property (the small boundary property). I will discuss a body of work related to these ideas.

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.