Topological Quantum Field Theories (TQFTs) have been studied as mathematical toy models for quantum field theories in physics and are described by a functor out of some bordism category. In dimension 2, TQFTs are fully classified by Frobenius algebras. Homotopy Quantum Field Theories (HQFTs), introduced by Turaev, consider additional homotopy data to some target space X on the bordism categories. For homotopy 1-types Turaev also gives a classification via crossed G-Frobenius algebras, where G denotes the fundamental group of X.
In this talk we will introduce a multi-object generalization of Frobenius algebras called Frobenius categories and give a version of this classification theorem involving the fundamental groupoid. Further, we will give a classification theorem for HQFTs with target homotopy 2-types by considering crossed modules (joint work with Alexis Virelizier).
 

Please attend if you would like to give a talk in the Logic Advanced Class this term.

Multiparameter persistence is an area of topological data analysis that synthesises the geometric information of a topological space via filtered homology. Given a topological space and a function on it, one can consider a filtration given by the sublevel sets of the space induced by the function and then take the homology of such filtration. In the case when the filtering function assumes values in the real plane, the homological features of the filtered object can be recovered through a "curved" grid on the plane called the extended Pareto grid of the function. In this talk, we explore how the computation of the biparameter matching distance between regular filtering functions on a regular manifold depends on the extended Pareto grid of these functions.