In this paper we provide an introduction to the Frank-Wolfe algorithm, a method for smooth convex optimization in the presence of (relatively) complicated constraints. We will present the algorithm, introduce key concepts, and establish important baseline results, such as e.g., primal and dual convergence. We will also discuss some of its properties, present a new adaptive step...
Multiscale high-contrast media can cause astonishing wave propagation phenomena through resonance effects. For instance, waves could be exponentially damped independent of the incident angle or waves could be re-focused as through a lense. In this review article, we discuss the numerical treatment of wave propagation through multiscale high-contrast media at the example of the...
Starting from the principle of locality in quantum field theory, which states that an object is influenced directly only by its immediate surroundings, we review some features of the notion of locality arising in physics and mathematics. We encode these in locality relations, given by symmetric binary relations, and locality morphisms, namely maps that factorise on products of...
This survey provides an overview of numerous results on $p$ -permutation modules and the closely related classes of endo-trivial, endo-permutation and endo- $p$ -permutation modules. These classes of modules play an important role in the representation theory of finite groups. For example, they are important building blocks used to understand and parametrise several kinds of...
We present a simple proof of the surface classification theorem using normal curves. This proof is analogous to Kneser’s and Milnor’s proof of the existence and uniqueness of the prime decomposition of 3-manifolds. In particular, we do not need any invariants from algebraic topology to distinguish surfaces.
On the occasion of Sir Roger Penrose’s 2020 Nobel Prize in Physics, we review the singularity theorems of General Relativity, as well as their recent extension to Lorentzian metrics of low regularity. The latter is motivated by the quest to explore the nature of the singularities predicted by the classical theorems. Aiming at the more mathematically minded reader, we give a...
This is an account on the theory of formal power series developed entirely without any analytic machinery. Combining ideas from various authors we are able to prove Newton’s binomial theorem, Jacobi’s triple product, the Rogers–Ramanujan identities and many other prominent results. We apply these methods to derive several combinatorial theorems including Ramanujan’s partition...
We present the lives and the work of four topologists who contributed to establish a center of topology at Heidelberg: Herbert Seifert, William Threlfall, Albrecht Dold and Dieter Puppe.
In this survey, we explain a few key ideas of the theory of graphs, and how these ideas have grown to form the foundation of entire research areas. Graph Theory is a fairly young mathematical discipline; here we explain some of its major challenges for the 21st century. László Lovász was recently awarded the Abel Prize. He made important contributions to all the areas discussed...
The Modular Isomorphism Problem asks if an isomorphism of group algebras of two finite $p$ -groups $G$ and $H$ over a field of characteristic $p$ , implies an isomorphism of the groups $G$ and $H$ . We survey the history of the problem, explain strategies which were developed to study it and present the recent negative solution of the problem. The problem is also compared to...
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...
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...
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...