Mathematical Colloquium

The Classification of Finite Simple Groups (CFSG), also known as the enormous theorem, is a highlight of 20th-century mathematics, both with respect to its mathematical content and to the complex process of proving the result. We will give a brief mathematical introduction and then focus on historical and cultural aspects, such as: changing perceptions of what a mathematical proof is, the character and the many contexts of mathematics as an intergenerational and international collaborative enterprise, the roles that trust and consensus play within this enterprise, and financial funding.

How much does modular arithmetic (i.e., calculating modulo $p$) tell us about the integers and the rational numbers? Under which circumstances can we use insights about fields of positive characteristic (e.g., finite fields, function fields over finite fields, or power series fields over finite fields) to understand fields of characteristic 0 (and conversely)?
Classical methods to transfer results between fields of different characteristics are the Lefschetz principle and the Ax-Kochen/Ershov Theorem which states that asymptotically, the theory of the $p$-adic numbers $\mathbb{Q}_p$ and of power series fields $\mathbb{F}_p((t))$ coincide. Tilting perfectoid fields, a recent approach developed by Scholze, gives a transfer principle between certain henselian fields of mixed characteristic and their positive characteristic counterparts and vice versa. In this talk, we survey various transfer principles and present a model-theoretic approach to tilting via ultraproducts, which allows us to transfer many first-order properties between a perfectoid field and its tilt. A key ingredient in our approach is an Ax-Kochen/Ershov principle for perfectoid fields (and generalizations thereof). This is joint work with Konstantinos Kartas (Sorbonne Université).

In this talk we focus on the construction of modern structure preserving methods for the simulation of (mainly) hyperbolic nonlinear conservation laws, for instance used to describe fluid dynamics. Our methodology of choice are high order accurate polynomial schemes, namely the discontinuous Galerkin (DG) framework. Our goal is to preserve inherent structural properties of the given hyperbolic PDEs such as preservation of kinetic energy and conservation/dissipation of entropy. Kinetic energy is an important quantity for the description of the behaviour of turbulence. Entropy on the hand is an important quantity that controls the physicallity of the energy transfer processes simulated.
It turns out that these structural properties are strongly related to the validity of simple product rules and chain rules of nonlinear functions. While these rules hold in the continuous case, i.e., for continuous smooth functions and derivative operators, in general, product and chain rules are not guaranteed anymore in the discrete case when using integration points, quadrature rules, discrete derivative operators, etc.
With the insight about the importance of product and chain rules equipped, we show how to use ingredients from finite difference, finite volume, and finite elements to construct specific discrete integration and differentiation operators that satisfy the specific product and/or chain rule properties. These discrete operators are the basis to construct full numerical methods on three dimensional unstructured curvilinear grids that discretetely preserve these structural properties.
We will also illustrate the benefit of such methods for applications, e.g., when simulating three dimensional turbulence.

Interfaces separating two or more species or components of a material are omnipresent in applications in the sciences. Nowadays there two main classes of models to describe interfaces, both from a theoretical and practical point of view: In classical so-called "sharp interface models" the species or components under consideration fill disjoint domains that are separated by lower dimensional surfaces of a certain regularity. On the other hand in "diffuse interface models" a partial mixing of the species or components on a small length scale is taken into account, which leads to an interfacial layer of small but positive thickness. This has the advantage that the interfaces do not need to be resolved explicitly and singularities in the interfaces can be described consistently. In this talk we will give an overview of some basic diffuse interface models with applications to material sciences and fluid mechanics. Moreover, we will discuss some analytic results on the relation between diffuse and sharp interface models, when the interfacial thickness of the diffuse interface tends to zero.