Mathematical Colloquium

Euler's beautiful formula for the values of the Riemann zeta function at the positive even integers can be seen as the starting point of the investigation of special L-values. Since then the field has developed widely and the investigation of special values of L-functions is nowadays a central area of modern number theory and arithmetic geometry. In this talk, I would like to give a gentle introduction to this topic for a general mathematical audience. I will mainly focus on the example of the Riemann zeta function. At the end of the talk, I will explain some aspects of special L-values related to my own research interests.

In the past, several polytope constructions arising from graphs have led to new insights in algebra, geometry, combinatorics and graph theory. In this talk, we will focus on so-called symmetric edge polytopes. After reviewing fundamental properties of these polytopes, such as reflexivity, central symmetry and their facet description, we will study regular unimodular triangulations of these polytopes. In particular, we are interested in the number of faces of a fixed dimension in such a triangulation and in the case of edges, we will derive sharp lower bounds, confirming conjectures by Ohsugi, Tsuchiya, Lutz and Nevo. Moreover, on the probabilistic side, we show that strengthenings of these conjectures are true asymptotically almost surely if one considers Erdös-Rényi graphs (in the subcrititcal and the supercritical regime). This is joint work with Alessio D'Alí, Daniel Köhne and Lorenzo Venturello. I will not assume any prior knowledge of any of the above mentioned topic.

A growing number of mathematicians are having fun explaining mathematics to computers using proof assistant softwares. This process is called formalization. In this talk, I'll describe what formalization looks like, what kind of things it teaches us, and how it could even turn out to be useful (in our usual sense of "useful"). This will not be a talk about the foundations of mathematics, and I won't assume any prior knowledge about formalization.

The automatic continuity problem asks the following question: Given two topological groups $L$ and $G$ and an algebraic morphism $\varphi\colon L\to G$, can we find conditions on the groups $L$ and $G$ ensuring that $\varphi$ is continuous? In this talk we will give an introduction to this problem focusing on the case where $L$ is a locally compact Hausdorff group and $G$ is a geometric group.

Originating with the work of Newton, there is a rich history of interactions between geometry of polyhedra and power series solutions to polynomial equations. In fact, a unifying principle behind these connections is provided by non-archimedean geometry. We shall present a panorama of some of the most salient developments in this direction including some quite recent results.