This is the first in a series of three courses on discrete geometry. The aim of the course is a skillful handling of discrete geometric structures including analysis and proof techniques. The material will be a selection of the following topics:

• Combinatorial geometry / Geometric combinatorics
• configurations of points, hyperplanes, subspaces
• Arrangements of points and lines, Sylvester-Gallai, Erdős-Szekeres
• Szemerédi-Trotter
• polyhedra and polyhedral complexes
• Polytope theory
• polarity, simple/simplicial polytopes
• shellability, face lattices, f-vectors, Euler and Dehn-Sommerville
• Subdivisions and triangulations (including Delaunay and Voronoi)
• Representations and the theorem of Minkowski-Weyl
• graphs, diameters, Hirsch (ex-)conjecture
• Arrangements, zonotopes, zonotopal tilings, oriented matroids
• regular polytopes, centrally symmetric polytopes
• extremal polytopes, cyclic/neighborly polytopes, stacked polytopes
• combinatorial optimization and 0/1-polytopes

For students with an interest in discrete mathematics and geometry, this is the starting point to specialize in discrete geometry. The topics addressed in the course supplement and deepen the understanding for discrete-geometric structures appearing in differential geometry, topology, combinatorics, and algebraic geometry.

Literature

• G. Ewald: "Combinatorial Convexity and Algebraic Geometry"
• P. McMullen, G.C. Shephard: "Convex Polytopes and the Upper Bound Conjecture"
• G.M. Ziegler: "Lectures on Polytopes"
• J. Matousek: "Lectures on Discrete Geometry"
• Further literature will be announced in class.

Prerequisites

Good knowledge of linear algebra. Some knowledge about combinatorics and geometry is helpful.