Michael Hartz, Department of Pure Mathematics and Michael Szestopalow, Combinatorics Optimization
“Brouwer and Sperner’s Excellent Adventure”
We will discuss Brouwer’s Fixed-Point Theorem and Sperner’s Lemma. Brouwer’s Fixed-Point Theorem is a fundamental result in analysis that has applications in all areas of mathematics. It asserts that every continuous map from a compact, convex subset of Rn into itself has a fixed point. Sperner’s Lemma states that if we take a triangulated simplex and colour the vertices in a special way, then there is a simplex of the triangulation whose vertices have distinct colours. Remarkably, these two results are equivalent. Michael will examine Brouwer’s result and some applications of fixed- point theorems, and Michael will highlight the (possibly) surprising connection between Brouwer and Sperner.