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Dept. of Mathematics, Ramakrishna Mission Vivekananda Educational and Research Institute (RKMVERI) has organised an colloquium on “Combinatorial applications of some topological theorems” by Professor R.B. Bapat, ISI, Delhi on 15 July 2021.

Title: Combinatorial applications of some topological theorems

Abstract

The Brouwer’s Fixed Point Theorem and the Borsuk-Ulam Theorem are important theorems in Topology with many applications. Brouwer’s Fixed Point Theorem asserts that a continuous function from a compact convex set to itself has a fixed point. We will describe equivalent versions of the theorem including Sperner’s lemma which is purely combinatorial. The Borsuk-Ulam Theorem states that any continuous function from the n-dimensional sphere to the n-dimensional euclidean space must map two antipodal points to the same point. Lovasz used the theorem to settle a long-standing conjecture about the chromatic number of a certain graph, called the Kneser graph. We will present this proof after introducing the necessary background.