Differential geometry, mathematical physics, and gauge theories : an honors thesis (HONRS 499)

Abstract

Modern physics relies heavily on geometric modeling of reality. It can be effectively phrased in terms of differential forms, vector fields, and other objects of differential geometry, and manipulated with the tools of differential geometry. Differential geometry itself relies on the development of advanced linear algebra, including the theory of tensor spaces and forms over vector spaces. Therefore, I develop advanced linear algebra, aspects of manifold theory, elements of pseudo-Riemannian geometry, a brief introduction to Lie groups, and some statements regarding vector bundles, in order to convey some aspects of modern physical theories, including the classical theory of electrodynamics, the theories of special and general relativity, and some notions of gauge theories.