MATH 7580: Topological Methods in Combinatorics (Spring 2010)

Instructor: Eran Nevo

A beautiful example of the course title is Lovasz proof of Kneser conjecture, determining the (large) chromatic number of certain graphs. The proof essentially uses the Borsuk-Ulam theorem in topology!

We will present several topological tools, and apply them to problems in combinatorics.

Our topological tools will include: fixed point theorems (e.g. Brouwer's), antipodality theorems (e.g. Borsuk-Ulam's) and generalizations using other group actions and equivariant cohomology, a bit of characteristic classes (namely Van Kampen obstruction), and discrete Morse theory.

Our combinatorial applications, will include: HEX game, graph chromatic numbers, Tverberg partitions (in convexity theory), (non)embeddability of simplicial complexes (e.g. Van Kampen - Flores complexes), homotopy type and enumerative properties of some simplicial complexes (e.g. independent-sets complexes and graph complexes), and more.