• Senior Thesis
• Department of Mathematics
• Harvey Mudd College

Connor Ahlbach

Harvey Mudd College Mathematics 2013

Thesis Proposal:	Combinatorial Analog of the Poincare Miranda Theorem
Thesis Advisor:	Prof. Francis Su
Second Reader:	Prof. Michael Orrison
E-Mail:	cahlbach@g.hmc.edu

A Discrete Approach to the Poincare Miranda Theorem

The Poincare-Miranda Theorem is a topological result about the existence of a zero of a function under particular boundary conditions. In this thesis, we explore proofs of the Poincare-Miranda Theorem that are discrete in nature - that is, they prove a continuous result using an intermediate lemma about discrete objects. We explain a proof by Tkacz and Turzanski that proves the Poincare-Miranda theorem via the Steinhaus Chessboard Theorem, involving colorings of partitions of n-dimensional cubes. Then, we develop a new proof of the Poincare-Miranda Theorem that relies on a polytopal generalization of Sperner's Lemma of Deloera - Peterson - Su. Finally, we extend these discrete ideas to attempt to prove the existence of a zero with the boundary condition of Morales.