Postdoctoral T.H. Hildebrandt Assistant Professor

Department of Mathematics

University of Michigan

**Colorful phenomena in discrete geometry and combinatorics via topological methods**

We will discuss two recent topological results and their applications to several different problems in discrete geometry and combinatorics involving colorful settings.

The first result is a polytopal-colorful generalization of the topological KKMS theorem due to Shapley. We apply this theorem to prove a new colorful extension of the d-interval theorem of Tardos and Kaiser, as well as to provide a new proof to the celebrated colorful Caratheodory theorem due to Barany. Our theorem can be also applied to questions regarding fair-division of goods (e.g., multiple cakes) among a set of players. This is a joint work with Florian Frick.

The second result is a new topological lemma that is reminiscent of Sperner’s lemma: instead of restricting the labels that can appear on each face of the simplex, our lemma considers labelings that enjoy a certain symmetry on the boundary of the simplex. We apply this to prove that the well-known envy-free division theorem of a cake is true even if the players are not assumed to refer non-empty pieces, whenever the number of players is prime or equal to 4. This is joint with Frederic Meunier.