Discrete Geometry I
The course focuses on basic notions and techniques in the field of Discrete Geometry, regarding point configurations and polytopes. The techniques include algebraic, topological, geometric and combinatorial methods. Details on the selected topics appear below.
- Radon's lemma, Helly's theorem, centerpoints, colorful Caratheodory theorem
- Euler's formula, crossing numbers, amplifcation through probabilistic method, Szemeredi-Trotter theorem, applications to sum-product estimates
- Unit distances problem, distinct distances, Erdos-Szekeres theorem via hypergraph Ramsey theory
- Number of joints via polynomial method
- Polytopes and polyhedra, Minkowski-Weyl theorem, Steinitz' theorem
- Balinksi's theorem, Hirsch conjecture, vertex-decomposibility
- Gale duality, non-rational polytopes, oriented matroids and their realizability
- Neighborly, cyclic, stacked polytopes, f-vectors, Dehn-Sommerville relations, shellability, upper bound theorem
- Triangulations, Voronoi and Delaunay, the associahedron.
Other or additional topics may be studied.
This course will be taught by Prof. Dr. Christian Haase (FU Berlin), Prof. Dr. Eran Nevo (HUJI), and Prof. Dr. Florian Frick (FU Berlin and Carnegie Mellon University).