9:30 AM — 10:30 AM
Speaker: Aaron Lauda, USC Dornsife
Location: McDonnell A10

Nonsemisimple Topological Quantum Computation

Since the foundational work of Freedman, Kitaev, Larsen, and Wang, it has been understood that 3-dimensional topological quantum field theories (TQFTs), described via modular tensor categories, provide a universal model for fault-tolerant topological quantum computation. These TQFTs, derived from quantum groups at roots of unity, achieve modularity by semisimplifying their representation categories—discarding objects with quantum trace zero. The resulting semisimple categories describe anyons whose braiding enables robust quantum computation.

This talk explores recent advances in low-dimensional topology, focusing on the use of nonsemisimple categories that retain quantum trace zero objects to construct new TQFTs. These nonsemisimple TQFTs surpass their semisimple counterparts, distinguishing topological features inaccessible to the latter. For physical applications, unitarity is essential, ensuring Hom spaces form Hilbert spaces. We present joint work with Nathan Geer, Bertrand Patureau-Mirand, and Joshua Sussan, where nonsemisimple TQFTs are equipped with a Hermitian structure. This framework introduces Hilbert spaces with possibly indefinite metrics, presenting new challenges.

We further discuss collaborative work with Sung Kim, Filippo Iulianelli, and Sussan, demonstrating that nonsemisimple TQFTs enable universal quantum computation at roots of unity where semisimple theories fail. Specifically, we show how Ising anyons within this framework achieve universality through braiding alone. The resulting braiding operations are deeply connected to the Lawrence-Krammer-Bigelow representations, with the Hermitian structure providing a nondegenerate inner product grounded in quantum algebra.

11:00 AM — 12:00 PM
Speaker: Francesco Lin, Columbia University 
Location: McDonnell A02

On integral rigidity in Seiberg-Witten theory

We introduce a framework to prove integral rigidity results for the Seiberg-Witten invariants of a closed 4-manifold X containing a non-separating hypersurface Y satisfying suitable (chain-level) Floer theoretic conditions. As a simple application, we show that if X has the homology of a four-torus, and it contains a non-separating three-torus, then the sum of all Seiberg-Witten invariants of X is determined in purely cohomological terms. Our results can be interpreted as (3+1)-dimensional versions of Donaldson's TQFT approach to the formula of Meng-Taubes, and build upon a subtle interplay between irreducible solutions to the Seiberg-Witten equations on X and reducible ones on Y and its complement. This is joint work with Mike Miller Eismeier.

1:15 PM — 2:15 PM
Speaker: Ian Zemke, University of Oregon
Location: McDonnell A02

An equivalence between two bordered categories

We describe an equivalence of categories between two natural bordered categories. The first consists of a subcategory of modules over the Lipshitz-Ozsvath-Thurston torus algebra. The second consists of a subcategory of modules over an algebra which encodes the Heegaard Floer Dehn surgery formulas. We will discuss the functors which appear, and describe some interpretations in terms of immersed curves. This is joint work with J. Hanselman and A.S. Levine.

2:30 PM — 3:30 PM
Speaker: Kristen Hendricks, Rutgers University
Location: McDonnell A02

Symplectic annular Khovanov homology and knot symmetry

Khovanov homology is a combinatorially-defined invariant which has proved to contain a wealth of geometric information. In 2006 Seidel and Smith introduced a candidate analog of the theory in Lagrangian Floer analog cohomology, which has been shown by Abouzaid and Smith to be isomorphic to the original theory over fields of characteristic zero. The relationship between the theories is still unknown over other fields. In 2010 Seidel and Smith showed there is a spectral sequence relating the symplectic Khovanov homology of a two-periodic knot to the symplectic Khovanov homology of its quotient; in contrast, in 2018 Stoffregen and Zhang used the Khovanov homotopy type to show that there is a spectral sequence from the combinatorial Khovanov homology of a two-periodic knot to the annular Khovanov homology of its quotient. (An alternate proof of this result was subsequently given by Borodzik, Politarczyk, and Silvero.) These results necessarily use coefficients in the field of two elements. This inspired investigations of Mak and Seidel into an annular version of symplectic Khovanov homology, which they defined over characteristic zero. In this talk we introduce a new, conceptually straightforward, formulation of symplectic annular Khovanov homology, defined over any field. Using this theory, we show how to recover the Stoffregen-Zhang spectral sequence on the symplectic side. This is joint work with Cheuk Yu Mak and Sriram Raghunath.