Inhalt:

This is the second in a series of three courses on discrete geometry. The aim of the course is a skillful handling of discrete geometric structures with an emphasis on metric and convex geometric properties. In the course we will develop central themes in metric and convex geometry including proof techniques and applications to other areas in mathematics.

The material will be a selection of the following topics:
Linear programming and some applications

• Linear programming and duality
• Pivot rules and the diameter of polytopes

Subdivisions and triangulations

• Delaunay and Voronoi
• Delaunay triangulations and inscribable polytopes
• Weighted Voronoi diagrams and optimal transport

Basic structures in discrete geometry

Basic structures in convex geometry

Volumes and roundness

Geometric inequalities

Geometry of numbers

Sphere packings

Applications in optimization, number theory, algebra, algebraic geometry, and functional analysis

• point configurations and arrangements
• incidence problems
• geometric selection theorems
  • epsilon-nets
  • convexity and separation theorems
  • convex bodies and polytopes/polyhedra
  • polarity
  • Mahler’s conjecture
  • approximation by polytopes
  • Hilbert’s third problem
  • volumes and mixed volumes
  • volume computations and estimates
  • Löwner-John ellipsoids and roundness
  • valuations
  • Brunn-Minkowski and Alexandrov-Fenchel inequality
  • isoperimetric inequalities
  • measure concentration and phenomena in high-dimensions
  • lattices
  • Minkowski's (first) theorem
  • successive minima
  • lattice points in convex bodies and Ehrhart's theorem
  • Ehrhart-Macdonald reciprocity
  • lattice packings and coverings
  • the Theorem of Minkowski-Hlawka
  • analytic methods