Sperner's Lemma: old and new proofs, generalizations, and surprising applications

Tuesday, 2-April-2024 

Abstract

Who doesn't like one of these three: geometry, topology, and combinatorics? And even if you don't, you will still love Sperner's lemma, which is a combinatorial statement that is equivalent to the Brouwer fixed point theorem in topology. I'll explain what it is, why it's so amazing, give heartwarming old and new proofs, and present a recent generalization to polytopes that has surprised me with diverse applications: to the study of triangulations, to fair division problems, and the Game of Hex.