Kolloquium Wilhelm Killing

In the context of fairly simple elliptic partial differential equations, overdetermined boundary conditions arise in the search of optimal shapes in a broad range of problems, e.g., in fluid mechanics, the theory of elasticity, electrostatics and integral geometry. Due to their relevance, the resulting overdetermined boundary value problems are addressed in prominent conjectures. The Berestycki-Caffarelli-Nirenberg conjecture from 1997, disproved by Sicbaldi in 2010, has lead to various recent results on the existence and classification of extremal unbounded domains. These unbounded optimal shapes can be regarded as analogues of constant mean curvature surfaces governed by nonlocal effects. Schiffer?s conjecture, and the related Pompeiu problem in integral geometry from 1929, are still open. In my talk, I will discuss a choice of classical and recent results on overdetermined boundary value problems, including joint work with M.M. Fall and I.A. Minlend.

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 billiards has also a curious link to basic geometric approximation theory. Then we will see polygons on which symplectic billiards has 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.

In this talk we introduce modular forms and harmonic weak Maass forms, real-analytic generalizations of holomorphic modular forms. We present applications of the theory in number theory and in the theory of elliptic curves.

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