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I will discuss joint work with Roman Bezrukavnikov on a categorical version of Hikita duality, which relates coherent sheaves on a symplectic resolution to constructible sheaves on the loop space of the dual resolution. I will focus on a basic case, where this can be made very explicit, and finish with some wild speculation on further generalisations.

I will discuss a theorem, joint work in progress with Constantin Teleman, in which we characterize which topological 3-dimensional Chern-Simons theories admit nonzero boundary theories.  Accepting some physical heuristics, it tells which gapped systems in 2+1 dimensions admit only gapless boundary systems.

I will discuss recent developments in describing the chiral algebras associated to 4d N=2 theories introduced by Beem et al. in terms of Omega backgrounds, and give a description of the class S chiral algebras following this perspective, in terms of boundary conditions, interfaces, and junctions in 4d N=4 SYM.

Then, I will present work in progress on a general TFT-type procedure for calculating the factorization algebras describing 2d CFTs which arise as compactifications of such configurations. I will show that this method correctly computes the class S chiral algebras, matching the construction of Arakawa, and discuss potential applications to computing the vertex algebras associated to toric divisors in toric CY3s, following Gaiotto-Rapcak.

 In the 90s Turaev, Viro, Barrett and Westbury constructed a (2+1)D state sum TQFT associated to any fusion category. The associated phases of matter were popularized by Kiteav, Levin and Wen and are now central  examples in condensed matter physics and quantum information theory.

Despite the importance of these phases, many of the computational techniques for working with fusion categories have not percolated into condensed matter physics. Many of these techniques are "folk theorems"
and have not appeared in the literature in a digestible form.

Jacob Bridgeman and myself have spent the last two years extracting these computational techniques from Corey Jones, and have been using them to understand defects in Levin-Wen phases. Our papers 1806.01279,
1810.09469, 1901.08069, 1907.06692 document the development of our understanding, and demonstrate how to do physically relevant fusion category computations.