Axiomatic Topological Quantum Field Theory

This dissertation provides an introduction to the ideas employed in topological quantum field theory. We illustrate how the field began by considering knot invariants of three-manifolds and demonstrating the consequences of defining such a theory axiomatically. The ideas of category theory are introduced and we show that what we are actually concerned with are symmetric monoidal functors from the category of topological cobordisms to the category of vector spaces. In two dimensions, these theories are elegantly classified through their equivalence to Frobenius algebras. Finally, we discuss possibly the simplest topological quantum field theory using a finite gauge group to define the fields.