MATH 7620 - Selected Topics in Discrete and Metric Geometry

Florian Frick, spring 2017.

Discrete geometry studies the geometry of discrete sets such as finite point sets in Euclidean space. Metric geometry is concerned with the geometry of metric spaces that are more general than Riemannian manifolds, where one still has a notion of curvature. These two branches of geometry are not entirely distinct, but rather metric geometry is an important tool in the discrete setting. We will explore some of the highlights of discrete and metric geometry. The topics entirely depend on the interests of the audience.

Possible topics include:

• polytope theory: CAT(c) spaces and diameters of polytopes, the Hirsch conjecture, Shepherd's conjecture on unfolding polytopes, existence of non-rational polytopes, upper and lower bound theorems
• triangulations of manifolds and point sets
• Voronoi diagrams and Delaunay triangulations
• Alexandrov spaces
• transversal theorems: Caratheodory, Helly, and Tverberg theorems
• Erdős distinct distances problem
• Čech and Vietoris--Rips complexes and their appearance in topological data analysis