Shadows in the Wild - Folded Galleries and Their Applications

This survey is about combinatorial objects related to reflection groups and their applications in representation theory and arithmetic geometry. Coxeter groups and folded galleries in Coxeter complexes are introduced in detail and illustrated by examples. Further it is explained how they relate to retractions in Bruhat-Tits buildings and to the geometry of affine flag varieties and affine Grassmannians. The goal is to make these topics accessible to a wide audience.

Continuation and Bifurcation in Nonlinear PDEs – Algorithms, Applications, and Experiments

Numerical continuation and bifurcation methods can be used to explore the set of steady and time–periodic solutions of parameter dependent nonlinear ODEs or PDEs. For PDEs, a basic idea is to first convert the PDE into a system of algebraic equations or ODEs via a spatial discretization. However, the large class of possible PDE bifurcation problems makes developing a general and user–friendly software a challenge, and the often needed large number of degrees of freedom, and the typically large set of solutions, often require adapted methods. Here we review some of these methods, and illustrate the approach by application of the package to some advanced pattern formation problems, including the interaction of Hopf and Turing modes, patterns on disks, and an experimental setting of dead core pattern formation.

Fitting Ideals in Number Theory and Arithmetic

We describe classical and recent results concerning the structure of class groups of number fields as modules over the Galois group. When presenting more modern developments, we can only hint at the much broader context and the very powerful general techniques that are involved, but we endeavour to give complete statements or at least examples where feasible. The timeline goes from a classical result proved in 1890 (Stickelberger’s Theorem) to a recent (2020) breakthrough: the proof of the Brumer-Stark conjecture by Dasgupta and Kakde.