Summer/Winter Schools

VSM Summer School 2023
September 10 to 16, 2023 in Dienten am Hochkönig
The School takes place at the Universitätssport- und Seminarzentrum Dientnerhof.
The program will start on Monday, September 11 in the morning and
finish on Friday, September 15 in the afternoon. Participants are
expected to arrive on Sunday, September 10 and to stay until Saturday,
September 16. Registration is open until July 31. Registration is open for all student members of the VSM.

Stefanie Sonner from Radboud University, Uli Wagner from IST Austria, and Wadim Zudilin from Radboud University
have already agreed to deliver classes. Please find further details below:

Stefanie Sonner from Radboud University
Degenerate Diffusion Equations and Applications in the Modelling of Biofilms
In this course I will give an introduction to degenerate diffusion equations motivated by models for biofilm growth. Biofilms are dense aggregations of bacterial cells attached to a surface and held together by a self-produced slimy matrix. They affect many aspects of human life and play a crucial role in natural, medical and industrial settings.

We will discuss important properties and mathematical difficulties of degenerate diffusion equations. To this end we will look at the porous medium equation as a prototype example and compare the qualitative behavior of solutions with the qualitative behavior of solutions of the classical heat equation. In particular, we will discuss self-similarity, finite speed of propagation, regularity and travelling wave solutions. Furthermore, we will connect the theory with applications looking at models for bacterial biofilms where degenerate diffusion effects play an important role.

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Uli Wagner from IST Austria
Topological Methods in Discrete Mathematics and Theoretical Computer Science
Methods and ideas from algebraic and geometric topology have been applied with great success (and often in very surprising ways) to prove results in combinatorics, discrete geometry, and theoretical computer science. A classical example is the following: What is the minimum number of colours necessary to color all the k-element subsets of an n-element set such that no pair of disjoint sets receive the same color (where n and k are positive integers with n>2k)? In 1978, Lovász used topological methods to prove that the answer is n-2k+2, resolving a conjecture of Kneser from 1955 that at first sight seemed to have no connection to topology.

In this course, we will give an introduction to this interplay between topology and discrete mathematics, which has seen rapid progress in recent years, illustrating it by a variety of topics and problems including topological lower bounds for graph coloring (of which Lovász‘ result is an example), decision-tree complexity and evasiveness of graph properties, and (non)embeddability and Tverberg-type results.

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Wadim Zudilin from Radboud University
A Rational Introduction to Irrationality
Though almost every real number happens to be irrational, proving that a particular – and a particularly interesting – number is, often requires very special techniques. This leads to numerous connections of number theory with other mathematics topics including asymptotic analysis, combinatorics of binomial sums, arithmetic of differential equations, tools from computer algebra and many more. The goal of my lectures is to highlight such links and their use in irrationality proofs for special numbers, like π and the values of Riemann’s zeta function at positive integers. We will start our journey with the classical Euclidean algorithm and continued fractions.