Summer School — TQFTs and their connections to representation theory and mathematical physics

TQFTs (Topological quantum field theories) are of central importance in several areas of mathematics and physics. From a mathematical point of view (going back to Atiyah), a TQFT is a symmetric monoidal functor from the cobordism category to some symmetric monoidal category. It can thus be seen as a representation of a fundamental geometric category on a target category and thereby organizes interesting algebraic structures, e.g. representations of mapping class groups, in terms of cobordism categories. In recent years, TQFTs have appeared in many more mathematical areas such as algebraic topology (the study of 4-manifolds) or algebraic geometry (the study of moduli spaces). In this Hausdorff school we would like to explore various facets of TQFTs related to representation theory and mathematical physics. The interplay of representation theory and TQFT is, at least, two-fold: on the one hand side, representation theory provides input data to construct TQFTs. The most famous example is given by the quantum groups Uq(sl(2)) from which a modular tensor category and thus a Chern-Simons type TQFT can be constructed. On the other hand, TQFTs explain and organize structures appearing in representation categories, for example character varieties and their quantizations.