Division Algebras, Global Forms of the Inverse Function Theorem, Differentiable Manifolds, and Fixed Point Theorems

Abstract

In this undergraduate thesis, we use results from Topology and Analysis, including but not limited to the Banach Fixed Point Theorem, in order to establish some global forms of the Inverse Function Theorem. As an application that brings together three different branches of mathematics, we prove a basic, yet important, result in Algebra: there is no commutative division algebra (not necessarily associative) that is isomorphic to a Euclidean space of dimension n for all n greater than or equal to 3. Furthermore, we will develop enough theory regarding differentiable manifolds to discuss and prove the Brouwer Fixed Point Theorem and the Schauder Fixed Point Theorem along with an application to game theory and economics. We will then compare and contrast the applications of the Banach Fixed Point Theorem and the Schauder Fixed Point Theorem to the field of differential equations. These applications are two famous theorems commonly known as the Picard-Lindelof Theorem and Peano's Theorem, respectively.