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Thomas Kerler
Topological Quantum Field Theory

Originally, the notion of Topological Quantum Field Theory (TQFT) was introduced by Atiyah and Witten in order to formalize properties of certain geometric field theories. In mathematics it has since evolved into what is better described as the highly structured representation theory of cobordisms, which has turned out to be particularly fruitful in dimension three. We will introduce the basic TQFT axioms as well as various refinements and generalizations that have naturally emerged in the descriptions of large families of mathematical TQFTs, especially in dim=3, and discuss their ramifications in numerous low-dim topological settings. Of particular interest are relations between geometrically (or "classically") constructed TQFTs with the combinatorially constructed TQFTs obtained using quantum groups and surgery presentations. We will discuss how integrality of the combinatorial TQFTs can be used to establish such connections.