Are mathematical proofs by computer legitimate? Already 50 years ago, Appel and Haken proved the century-old four-colour problem with thousands of case distinctions that could only reliably be analysed by computer. A very simple special case (no computer needed) is the three-colour theorem: a map where every country has an even number of neighbours can be coloured with three colours so that countries with a common boundary line have different colours. Planar maps are also the surfaces of polytopes such as cubes or tetrahedra, and when colouring their faces, their multi-coloured corners are a natural representation of so-called Nash equilibria in two-player games. This connection is the topic of this talk.

The three-colour theorem can be used to show that there are games with three strategies for one player (and any number of strategies for the other player) that have the maximum possible number of Nash equilibria. Games with more strategies give rise to polytopes in higher dimensions, and here we needed computer help to solve a 25-year-old problem on the maximum number of Nash equilibria of a game with five strategies on each side. An upper bound is 42, and for the corresponding “dual neighbourly” polytopes a full list of 159,375 combinatorial types can be checked with suitable obstructions that their maximum equilibrium number is 32. But further tricks are needed for more general polytopes where no such list exists to show that this is the true bound.

This research is joint work with Constantin Ickstadt and Thorsten Theobald (University of Frankfurt).

Lattice polytopes are a central subject of study in discrete geometry, with many applications to other fields like toric geometry or integer optimization. The integer decomposition property (IDP) is a niceness criterion which some polytopes fulfill and others don't. A unimodular triangulation, if it exists, guarantees IDP. We discuss unimodular triangulations of supposedly simple polytopes – parallelepipeds of dimensions three and four.

Periodic timetabling is a fundamental problem in public transport planning, commonly modeled by the Periodic Event Scheduling Problem (PESP). Despite its widespread use, solving PESP instances efficiently remains challenging due to their computational complexity. Recent results establish a geometric connection between feasible periodic timetables and integer points in certain zonotopes. Building on this perspective, this thesis studies the structure and fine zonotopal tilings of two closely related objects, the cographic zonotope and the cycle offset zonotope, and investigates how these structures can be exploited algorithmically. We characterize neighborhood relations between tiles and develop algorithms for identifying neighboring tiles, locating the tile containing a given integer point, and finding integer points within tiles. Combining these results with the cycle periodicity property of PESP leads to the main contribution of the thesis, the Zonotopal Tiling Search, a new heuristic optimization method for periodic timetabling. The method interprets timetable optimization as a local search over the cells of a fine zonotopal tiling, providing a novel geometric approach to finding locally optimal periodic timetables.

Popular films often portray mathematicians as isolated geniuses, while mathematics itself appears as a backdrop of impressive but unexplained formulas. How can cinema offer an alternative to this “mathematical misanthrope” stereotype?

The talk centers on Hidden Figures (2016), examining how its portrayal of Katherine Johnson challenges stereotypical mathematical worldviews and presents mathematics as a process involving sustained work, collaboration, and problem-solving. It connects this analysis with a closer look at numerical approximation through Euler’s method. A NASA technical report co-authored by Johnson provides a historical point of comparison. An accompanying short video essay developed as a science communication format brings contrasting cinematic representations into dialogue. Drawing on the AEIOU framework of science communication—Awareness, Enjoyment, Interest, Opinion formation, and Understanding—the talk explores how film can invite audiences to engage with mathematics beyond learning formulas. Together, these examples explore how film can challenge mathematical stereotypes and contribute to science communication, while distinguishing between making mathematical work meaningful and explaining a method in detail.