The current organizers are Christopher Dean and Andre Kornell. Information about fall 2023 seminars can be found here.
The functorial difference operator
As a tool for studying the structure of endofunctors F of Set, we introduce the difference operator △
△[F](X) = F(X + 1) \ F(X).
This is analogous to the classical difference operator for real valued functions, a discrete form of the derivative.
The \ above is set difference and can't be expected to be functorial, but it is for a large class of functors, the taut functors of Manes, which include polynomial functors and many more.
We obtain combinatorial versions of classical identities, often ''improved''. Many examples will be given.
The talk should be accessible to everyone. The only prerequisite is some very basic category theory.
QFTs in a stable universe
Thanks to work of Atiyah and Segal in the late '80s, it has been understood that quantum field theories furnish us with representations of bordism categories. In this talk applications of these ideas in stable categories will be discussed. I will review the basic definitions involved, then present classification results using the language of Thom spectra.
Tannaka-Krein Reconstruction for Fusion 2-categories
Tannaka-Krein reconstruction is a well known procedure which recovers a Hopf algebra from its category of modules and monoidal fiber functor. Following the approach of P. Schauenburg, we show how to recover a semisimple Hopf category from its fusion 2-category of representations. I will also provide some remarks on in-progress work on a more general reconstruction theorem applicable to any symmetric fusion 2-category.
Computing the TVBW 3-manifold invariants from Tambara-Yamagami categories
I'll give a quick intro to spherical fusion categories and the Turaev-Viro-Barrett-Westbury construction, which associates an invariant of oriented 3-dimensional manifolds to each such category. Some of the simplest spherical fusion categories are the so-called Tambara-Yamagami categories, which depend on the data of a finite abelian group A, a choice of isomorphism between A and its dual, and a sign +1 or -1. Despite their fairly simple definition, these categories are known to give rise to TVBW invariants that are NP-hard to compute. I'll explain what this means, and then describe my recent work with Colleen Delaney and Clement Maria establishing an efficient algorithm for computing these invariants on 3-manifolds with bounded first Betti number. I will also try to say a few things about why such complexity/algorithm results are interesting in the context of 3-manifold topology and quantum computation.
Axioms for the category of finite-dimensional Hilbert spaces and linear contractions
I will explain the motivation and main ideas behind recent joint work with Chris Heunen (arXiv:2401.06584) that characterises the category of finite-dimensional Hilbert spaces and linear contractions. The axioms are about simple category-theoretic structures and properties. In particular, they do not refer to norms, continuity, dimension, or real numbers. The proof is noteworthy for the new way that the scalars are identified as the real or complex numbers. Instead of resorting to Solèr’s theorem, which is an opaque result underpinning similar characterisations of other categories of Hilbert spaces, suprema of bounded increasing sequences of scalars are explicitly constructed using directed colimits of contractions. To keep the talk accessible, I will not assume prior familiarity with dagger categories, instead introducing the relevant concepts as needed.
Connections in algebraic geometry via tangent categories (part I)
Connections in algebraic geometry via tangent categories (part II)
Updated February 21, 2023.