Topological Combinatorics is a relatively new field of mathematics with many applications in mathematics and computer science. The field is concerned with an unexpected usefulness of topological methods (methods arising mostly from the study of continuous objects and deformations) to solving combinatorial problems (often dealing with discrete objects like graphs and colorings of graphs).
A striking example of this was the resolution of a conjecture by Kneser about the chromatic number of certain graphs (called Kneser graphs). Although combinatorial in nature, this problem was first solved by László Lovász as an application of methods from topology and in particular by associating with every Kneser graph a topological space that ends up being a high-dimensional sphere. Imre Bárány later gave another proof using the Borsum-Ulam theorem, a certain statement about continuous maps defined on spheres in arbitrary dimensions.
Another type of example where topology plays an unexpected role are fair-division problems, where some resource needs to be divided in "equal" parts among a certain number of people. See for example the ham sandwich theorem. Again, although combinatorial in nature, such problems can be resolved as an application of the Borsum-Ulam theorem.
Finally, the combinatorial statement known as Sperner's lemma turns out to be equivalent to the Brouwer's fixed-point theorem in topology.
In this seminar, we will explore this vast topic by understanding the aforementioned examples and as time permits use this as a motivation to dip a toe in the area of algebraic topology and homology theory.
Bibliography:
- A course in Topological Combinatorics, by Mark de Longueville
