# UCSB Quantum Algebra and Topology Seminar

## Fall 2018 Schedule - 11AM on Tuesdays in South Hall 6635.

#### Upcoming Talks:

November 13 | Nicolle E.S. Gonzalez (USC)

Categorical Bernstein Operators and the Boson-Fermion correspondence

Bernstein operators are vertex operators that create and annhilate Schur polynomials. These operators play a significant role in the mathematical formulation of the Boson-Fermion correspondence due to Kac and Frenkel. The role of this correspondence in mathematical physics has been widely studied as it bridges the actions of the infinite Heisenberg and Clifford algebras on Fock space. Cautis and Sussan conjectured a categorification of this correspondence within the framework of Khovanov's Heisenberg category. I will discuss how to categorify the Bernstein operators and settle the Cautis-Sussan conjecture, thus proving a categorical Boson-Fermion correspondence.

November 20 | Mikhail Khovanov (Columbia)

November 22 | No meeting (Thanksgiving)

Dec 4 | Paul Wedrich (ANU)

#### Past Talks:

October 30 | Modjtaba Shokrian Zini (UCSB)

Hopf monads and generalized symmetries of fusion categories

Hopf monads in monoidal categories are a generalization of the notion of categorical Hopf algebras which are defined only in braided categories. We will go over the definition, then explore the possible role they might play for a generalization of group symmetries of fusion categories. Indeed, it turns out that any group symmetry can be expressed as a special case of a Hopf monad symmetry. The ultimate goal is to derive the extension theory needed to study the classification of fusion categories. Likely, a similar extension theory should exist for the richer modular categories.

October 23 | Alex Turzillo (Caltech)

Diagrammatic State Sums for 2D Pin-Minus TQFTs

This talk will introduce a state sum construction of two dimensional pin-minus TQFTs based on the connection between pin-minus surfaces and immersions of unoriented surfaces into $$\mathbb{R}^3$$. In addition to some non-invertible theories, the construction produces all invertible pin-minus TQFTs, including the theory whose closed partition function is the Arf-Brown-Kervaire invariant.

October 9 | Yang Qiu (UCSB)

Representation of mapping class group from DW theory and related calculation

This talk will present the construction of untwisted Dijkgraaf Witten theory and representation of mapping class groups in a combinatorial way. A description for the state space and the action of the mapping class group will be derived. Some calculations for simple cases will be presented.

October 2 | Joe Moeller (UCR)

Categorical Network Theory

Network theory is a diverse subject which developed independently in several disciplines. It uses graphs with additional structure to model everything from complex systems to theories of fundamental physics. In this talk, we'll look at certain categorical perspectives on network theory which have been developed in recent years. In particular, we will discuss the theory of network models, a tool used to construct operads which capture the combinatorics of generalized networks.