Mathematical Colloquium

Problems with multiple spatial scales occur in many applications, for instance when considering modern composite materials. Numerically, such problems are challenging as standard discretization would lead to huge systems. Instead, computational multiscale methods construct their basis functions from local solutions of the PDE, thereby generating problem-adapted approximation spaces. While this yields good approximations already on coarse meshes and is very efficient if many right-hand sides need to be studied, the problem-adapted approach becomes more difficult when the multiscale coefficients change themselves. In this talk, we consider the Localized Decomposition Method (LOD) and discuss some recent ideas how to reduce this burden. As exemplary situation we concentrate on coefficients with random defects where we present a so-called offline-online strategy. We present a priori error estimates as well as illustrative numerical experiments.

Projective geometry is one of the oldest disciplines of mathematics. On the other hand, the revolutionary discovery of the Fargues-Fontaine curve is only a few years ago. In my talk I will give a qualitative comparison of this curve with the projective line and explain its significance for modern arithmetic geometry.

Tame geometry (or tame topology) is a large program to identify, study and classify geometries in which pathological phenoma like space-filling curves and counter-intuitive results like the Banach-Tarski paradox can never arise. Originally envisioned by Grothendieck as topologie modérée in his “Esquisse d’un Programme”, model theorists using syntactic tools from logic have proposed o-minimality as a candidate for the axiomatic approach proposed by Grothendieck. Indeed, o-minimal geometry, an already wide-ranging generalization of both semi-algebraic and sub-analytic geometry, provides an excellent framework for developing tame geometry, and has recently seen stunning success through applications in number theory related to the André-Oort conjecture. However, o-minimality is far from the only tame geometry arising from syntactical/logical considerations, and progress has been made in identifying and classifying such geometries. In this talk I review the basic model-theoretic/logical setup and o-minimality, and then provide further examples and classifications of tame geometries. No background knowledge in logic is assumed.

Just like the field of real numbers and the field of complex numbers, which are well-known and indispensable in many areas of mathematics, the field of $p$-adic numbers and the field of formal Laurent series over a finite field with $p$ elements are further examples of so-called local fields and play an important role in various branches of number theory. These two fields, although of different characteristic, are known to behave very similar in many ways, but often problems turn out easier or harder for one of the two. In this talk I will discuss specifically the algorithmic problem of deciding whether a given polynomial equation has a solution in the field, which for the field of $p$-adic numbers was solved already in the 60's, but for fields of Laurent series over finite fields was open for a long time. I will start with a gentle introduction to local fields and will then explain the approach to such decidability questions through the model theory of valued fields.