Schedule, Notes/Handouts & Videos

One consequence of the cobordism hypothesis is that there's a homotopy coherent action of O(n) on the space of fully dualizable objects in a symmetric monoidal n-category. But what does this mean concretely? For example, by joint work with Douglas and Schommer-Pries there's an O(3) action on the 3-groupoid of fusion categories, but how do you turn this abstract action into a concrete collection of statements about fusion categories? In this talk, I will explain joint work in progress with Douglas and Schommer-Pries answering this question by saying such an action is given by four explicit pieces of data. If time permits I will also discuss homotopy fixed points for SO(3) actions, and explain the relationship between homotopy fixed points and spherical structures on fusion categories.

Character varieties are moduli spaces of representations of fundamental groups of manifolds. In dimensions 2 and 3 these exhibit symplectic and Lagrangian structures respectively, and it is an important question in 3d and 4d TQFT to quantize these structures functorially. In the 90's and 00's, three quantization schemes were proposed: via so-called Alekseev-Grosse-Schomerus algebras, via skein theory, and via Fock-Goncharov cluster quantizations. The AGS and FG constructions were intrinsically algebraic, but lacked the desired topological functoriality, while the skein construction is intrinsically topological but algebraically inaccessible. It is by now understood how to obtain AGS algebras and skein theory in the context of factorization homology, in particular how to unify the two quantizations. In this talk I will explain joint work with Ian Le, Gus Schrader and Sasha Shapiro which constructs Fock-Goncharov cluster quantizations from factorization homology. An interesting new ingredient is that of parabolic induction domain walls labeling line defects on the surface.

A major open problem in quantum topology is the construction of an oriented 4-dimensional topological field theory in the sense of Atiyah-Segal which is sensitive to exotic smooth structure. In this talk, I will sketch a proof that no semisimple field theory can achieve this goal and that such field theories are only sensitive to the homotopy types of simply connected 4-manifolds. This applies to all currently known examples of oriented 4-dimensional TFTs valued in the category of vector spaces, including unitary field theories and once-extended field theories which assign algebras or linear categories to 2-manifolds. If time permits, I will give a concrete expression for the value of a semisimple TFT on a simply connected 4-manifold, explain how the presence of `emergent fermions’ in a field theory is related to its potential sensitivity to more than the homotopy type of a non-simply connected 4-manifold, and comment on implications for the theory of ribbon fusion categories. This is based on arXiv:2001.02288.

The Kuperberg invariant is a quantum invariant of 3-manifolds based on any finite dimensional Hopf algebras, which is also related to the Turaev-Viro-Barrett-Westbury invariant. We give a construction of an invariant of 4-manifolds that can be viewed as a 4-dimensional analog of the Kuperberg invariant. The algebraic data for the construction is what we call a Hopf triple which consists of three Hopf algebras and a bilinear form on each pair of the Hopf algebras satisfying certain compatibility conditions. Quasi-triangular Hopf algebras are special examples of Hopf triples. Trisection diagrams of 4-manifolds proposed by Gay and Kirby will be used in the construction of the invariant. Some interesting properties of the invariant will also be discussed.

In the recent study of quantum error correction codes, a class of new 3+1D models were discovered which host both features that are similar to the usual topological models and ones that are drastically different. On the one hand, similar to topological models, these so-called ‘fracton’ models have ground state degeneracy that is stable against any local perturbation. There are also fractional point excitations in the bulk, which cannot be created or destroyed on their own. On the other hand, the ground state degeneracy is not a topology dependent constant. Instead it grows with system size. At the same time, the fractional excitations have restricted motion, i.e. they cannot move freely in the full three dimensional space. It is obvious that the fracton models are beyond the framework of TQFT, but are also closely related to it in many important ways. In this talk, I will review some of the most interesting examples of fracton models and explain how we propose to interpret their nontrivial features as depending on not only the topology of the underlying manifold, but also its ‘foliation’ structure, where each foliation leaf carries a 2+1D TQFT of its own. While our understanding of the fracton models are very limited at this point, they are pointing to a way to generalize TQFT in higher dimensions.

Where are we with COVID-19, and how are mathematical models and statistics helping us develop strategies to overcome the burden of infections. Nicholas P. Jewell is Chair of Biostatistics and Epidemiology at the London School of Medicine and Tropical Medicine and Professor of the Graduate School (Biostatistics and Statistics) at the University of California, Berkeley.

Using factorization homology, I'll outline a conceptual construction of link invariants, generalizing the Jones and Reshetkhin-Turaev polynomial invariants, using factorization homology.  I'll explain how to supply a conceptual proof of the 1-dimensional tangle hypothesis, using essentially the same construction.  This is joint work with John Francis.

Let G denote a class of manifolds (such as SO (oriented), O (unoriented), Spin, Pin+, Pin-, manifolds with spin defects).  We define a 2+1-dimensional G-modular TQFT to be one which lives on the boundary of a bordism-invariant 3+1-dimensional G-TQFT.  Correspondingly, we define a G-modular braided category to be a G-premodular category which leads to a bordism-invariant 3+1-dimensional TQFT.  When G = SO, this reproduces the familiar Witten-Reshetikhin-Turaev TQFTs and corresponding modular tensor categories.  For other examples of G, non-zero G-bordism groups in dimensions 4 or lower lead to interesting complications (anomalies, mapping class group extensions, obstructions to defining the G-modular theory on all G-manifolds).