Algebraic techniques and applications to combinatorial problems, e.g. linear and exterior algebraic methods and intersection theorems; the combinatorial Nullstellensatz and graph coloring; Stanley-Reisner rings and face numbers of polytopes and simplicial complexes; algebraic constructions in extremal combinatorics.

Discrete geometry investigates combinatorial properties of configurations of geometric objects.The topics of this course include: basics on convex sets, convex polytopes, and hyperplane arrangements; combinatorial complexity of geometric configurations; intersection patterns and transversals of convex sets; geometric Ramsey-type results; polyhedral combinatorics and high-dimensional convexity.

We introduce the basics of Fourier analysis on finite abelian groups and discuss applications in Theoretical Computer Science and Combinatorics. These include: bounds for error correcting codes, threshold phenomena in random graphs, voting schemes and influences in Boolean functions, probabilistically checkable proofs, Fermat's Last Theorem over finite fields.

This seminar is held once a year and complements the course ``Approximate Methods in Geometry''. Students of the seminar will present original research papers on approximate methods, most of them very recent. The seminar is a good preparation for a master thesis in the area. In the Spring semester, we offer a similar seminar geared towards topics around the course ``Computational Geometry".

Elementary topological notions and results: simplicial & cell complexes, homotopy of maps, nerve theorem, Borsuk-Ulam-type theorems, connectivity, (deleted) joins & products, finite group actions and equivariant maps. Geometric & combinatorial applications: Ham-Sandwich & partition theorems, Kneser's conjecture, van Kampen-Flores-type theorems, topological & colored Tverberg theorem.