RESEARCH IN RESIDENCE — Ky Fan type theorems, their relatives and generalizations

Combinatorial statements, such as theorems of Caratheodory, Radon, Helly, Sperner, Tucker, Ky Fan, etc., are fundamental results of combinatorial (algebraic) topology, accessible to non-specialists, which are immediately applicable to mathematical economics, data science, game theory, graph theory, mathematical optimization, computational geometry, and other fields. The aim of our project is to study the “Ky Fan correspondence » between the topology of triangulated spherical bundles and combinatorics of generalized labelings (coloring) of the associated simplicial complexes. Guided by the review of De Loera et. al. (Bull. Amer. Math. Soc., 2019) we explore the consequences of this correspondence for the envy-free and fair division problem, Tverberg-type theorems, and other related problems of topological combinatorics.