Algebra/Topology seminar

Speaker: Renee Hoekzema

Title: Invertible and striped cylinder TQFTs

Abstract: Cobordism categories describe the gluing structure of manifolds, and functorial topological quantum field theories are their algebraic representations. Classifications of TQFTs tell us about the structure of manifolds as well being relevant to mathematical physics. In this talk I will discuss two very different new approaches to classifying variants of TQFTs. On the one hand, invertible TQFTs are classified by SKK (controlled cut and paste) manifold invariants. SKK groups are related to cobordism groups through a short exact sequence, and we resolve the splitting problem as well as the kernel for a wide range of tangential structures on manifolds, leading to a table of classifications of invertible (not necessarily unitary) TQFTs relevant for physics. On the other hand, we introduce substructure in the form of submanifolds to the cobordism categories, so-called nested cobordism categories, and consider the more intricate structure arising from nested gluing. The first example of a category of cylinders endowed with line cobordisms already gives rise to interesting algebraic structure related to Temperly-Lieb algebras.