Senior Thesis

Department of Mathematics
Harvey Mudd College

John Cloutier

Harvey Mudd College Mathematics 2003

Thesis: A Combinatorial Analogue of The Poincaré-Birkhoff Fixed Point Theorem

Thesis Advisor: Prof. Francis E. Su
Second Reader: Prof. Lesley A. Ward

A Combinatorial Analogue of the Poincaré-Birkhoff Fixed Point Theorem

Results from combinatorial topology have shown that certain combinatorial lemmas are equivalent to certain topologocal fixed point theorems. For example, Sperner's lemma about labelings of triangulated simplices is equivalent to the fixed point theorem of Brouwer. Moreover, since Sperner's lemma has a constructive proof, its equivalence to the Brouwer fixed point theorem provides a constructive method for actually finding the fixed points rather than just stating their existence. The goal of this research project is to develop a combinatorial analogue for the Poincaré-Birkhoff fixed point theorem.