AZ Ph.D. Seminar
|Friday, 2 June
Representation theory in the limit
Abstract: First, we are going to discuss perverse schobers (categorical generalisations of the concept of perverse sheaves). Secondly, we are going to discuss topological Fukaya categories.
|Friday, 9 June
Relative Serre functors and pivotality in categorical Morita theory
Abstract (from one of David's papers): The Morita context provided by an exact module category over a finite tensor category gives a two-object bicategory with duals. Right and left duals of objects in the module category are given by internal Homs and coHoms, respectively. We express the double duals in terms of relative Serre functors, which leads to a Radford isomorphism for module categories. There is a bicategorical version of the Radford S4 theorem: on the bicategory of a Morita context, the relative Serre functors assemble into a pseudo-functor, and the Radford isomorphisms furnish a trivialization of the square of this pseudo-functor, i.e. of the fourth power of the duals. We also show that the Morita bicategories coming from pivotal exact module categories are pivotal as bicategories, leading to the notion of pivotal Morita equivalence. This equivalence of tensor categories amounts to the equivalence of their bicategories of pivotal module categories.
Friday, 28 July
Derived Turaev-Viro Theories
Abstract: One of the most fundamental examples of topological field theories (Tfts) are those of Turaev-Viro type. These theories arise as fully extended Tfts induced by the datum of a separable semi-simple tensor category. A lot of work has been done in the last decade to drop the semi-simplicity assumption and to define 3-dimensional Tfts for more interesting non-semi-simple categories.
On the relationship between 2d CFTs and 3d TQFTs
Abstract: In my PhD project I am studying surface defects in finite non-semisimple three-dimensional topological quantum field theories with the goal of applying them in a holographic setting to describe properties of two-dimensional conformal field theories which appear as boundary theories of such 3d TQFTs.
|Thursday, 3 August
The orbifold machine - or: constructing MTCs from lattices using VOAs
Abstract: Modular tensor categories (MTCs) arise naturally in conformal field theory as categories of representation for vertex operator algebras (VOAs). After a short introduction to VOAs and their representation theory, I will sketch how VOAs can be used to construct interesting MTCs from well-understood ones (pointed MTCs) using the orbifold construction. Crossed categories, (de)equivariantisation, projective representation theory and Galois extensions arise naturally in this context.
A Classification of Derived TFTs
Abstract: According to the Cobordism Hypothesis, n-dimensional framed extended topological field theories are classified by fully dualizable objects in a specified symmetric monoidal (∞,n)-category C. For low n, and C the n-category of n-vector spaces, these objects can be characterized in simpler terms: In dimension 1 we obtain finite-dimensional vector spaces, for n=2 separable algebras, and for n=3 (separable) multifusion categories. Johnson-Freyd has sketched how a notion of multifusion n-categories extends this correspondence to higher dimensions. I apply the theory of enriched ∞-categories and their Cauchy-completions to formalize and generalize this. Notably, my methods allow for replacing the field k by an E_n-algebra in any presentably E_n-monoidal ∞-category; yielding for example a notion of derived multifusion (∞,n)-categories.