Lecture: Topological methods in Combinatorics (M. Balko and M. Tancer)

General information

Schedule and tutorials

Lecture is scheduled to the room S7 (Malá Strana) on Mondays, 12:20.
Tutorials are taught by Todor Antic. They are scheduled to the room S6 (Malá Strana) on Mondays 14:00. See the tutorial webpage.

Content of the lecture

The traditional content of Topological Methods in Combinatorics is a brief introduction to topology (intended for students of computer science who possibly does not meet it in other lectures) and then the main content are applications of topology in combinatorics, discrete mathematics or combinatorial geometry. (This should be new both for students of mathematics or computer science). One example of such an application is that the chromatic number of so-called Kneser graphs can be determined via topological tools.

Previous lectures

If you are interested in the content of the lectures Topological Methods in Combinatorics of Mathematics++ in previous years, you can check: Topological methods in 2022, or Mathematics++ in 2024.

Language

The language of the lecture will be Czech or English depending on the audience. (If there is an attendee who does not understand Czech, the lecture will be in English).

Literature (may be extended during the semester)

[M] J. Matou¹ek: Using the Borsuk-Ulam theorem, Springer 2003.
[H] Algebraic Topology by A. Hatcher: http://www.math.cornell.edu/~hatcher/ .
[Mu] J. R. Munkres: Elements of Algebraic Topology
Research articles:
[BMZ] P. Blagojeviæ, B. Matschke, G. M. Ziegler: Optimal bounds for the colored Tverberg problem
[MTW] J. Matou¹ek, M. Tancer, U. Wagner: A geometric proof of the colored Tverberg theorem

Topics Covered

Lecture	Contents	Literature
Lecture 1, 23.2. (MB)	Introduction. Kneseger's graph KG(n,k), chromatic number of Kneser's graphs (just statement), the Ham Sandwich Theorem. fair cutting of convex bodies. Basic definitions from general topology: topological space, open set and closed set, Hausdorff space, subspace, continous mapping, homeomorphism, closure, interior and boundary, compact sets, connectivity, path-connectivity. (Strong) deformation retract. Presentation [PDF],notes (in Czech) [PDF].	[M, Chap. 1].
Lecture 2, 2. 3. (MB)	Homotopic mappings, homotopy of topological spaces, contractible space. Geometric simplicial complex, carrier (polyhedron). Triangulation of the sphere as a boundary of a simplex and as a boundary of a crosspolytope. Abstract simplicial complex, geometric representations of an abstract simplicial complex. Continuous mapping between polyhedra of simplicial complexes derived from a simplicial mapping, statement of the Simplicial approximation theorem barycentric subdivisions. Presentation [PDF], notes (in Czech) [PDF].	[M, Chap. 1].
Lecture 3, 9. 3. (MB)	The Borsuk-Ulam Theorem, proof of all the equivalences. Proof of Brouwer's Fixed Point Theorem using the Borsuk-Ulam Theorem. Presentation [PDF], notes [PDF].	[M, Chap. 1].