Course Description

This is the first in a series of three courses on Discrete Geometry. We will get to know fascinating geometric structures like configurations of points and lines, hyperplane arrangements, and in particular polytopes and polyhedra, and learn how to handle them using modern methods for computation and visualization and current analysis and proof techniques. A lot of this looks quite simple and concrete at first sight (and some of it is), but it also very quickly touches topics of current research.

The material will be a selection of the following topics:

Basic structures in discrete geometry

• Polyhedra and polyhedral complexes
• Configurations of points, hyperplanes, subspaces
• Subdivisions and triangulations (including Delaunay and Voronoi)
• Examples and Problems

Combinatorial geometry / Geometric combinatorics

• Arrangements of points and lines, Sylvester-Gallai, Erdös-Szekeres,
• Szemeredi--Trotter
• Arrangements, zonotopes, zonotopal tilings, oriented matroids
• Examples and problems (challenge problem: simplicial line arrangements)

Polytope theory

• Representations and the theorem of Minkowski-Weyl
• Polarity, simple/simplicial polytopes
• Shellability, face lattices, f-vectors, Euler- and Dehn-Sommerville
• Graphs, diameters, and the Hirsch (ex-)conjecture

Examples, examples, examples

• regular polytopes, centrally symmetric polytopes
• extremal polytopes, cyclic/neighborly polytopes, stacked polytopes
• combinatorial optimization and 0/1-Polytope

Geometry of linear programming

• Linear programs, simplex algorithm, LP-duality

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 that appear in differential geometry, optimization, combinatorics, topology, and algebraic geometry.

Lectures

Contact