Topological Quantum Field Theory and the Representation Theory of Manifolds

The Department of Mathematics is pleased to welcome Lukas Mueller, for his talk titled, “Topological Quantum Field Theory and the Representation Theory of Manifolds.”

Abstract: Representation theory studies concrete realizations of abstract algebraic structures — such as groups — by linear operators acting on vector spaces. In a similar spirit, topological quantum field theories (TQFTs) can be viewed as a kind of “representation theory of manifolds,” where cutting and gluing operations play the role of composition. Originating in quantum physics, the subject has developed into a rich and independent area of mathematics.

TQFTs provide a powerful framework for uncovering surprising connections between low-dimensional topology, algebra, and higher category theory. In the first part of the talk, I will describe recent applications of these ideas that relate duality structures on categories to the topology of two-dimensional surfaces. I will explain how this perspective leads to new approaches to classical problems in the study of mapping class groups.

The analogy with representation theory — and with physics — suggests that unitary versions of TQFTs should play a central role in many applications. However, the higher-categorical structures needed to define and study unitary TQFTs have only begun to be understood in recent years. In the second part of the talk, I will introduce some of the emerging mathematical structures in this area and present first results.