### Fall 2018 Seminar

##### Organizers: Stephen Bigelow, Colleen Delaney, Eric Samperton

###### Back to current seminar schedule

Dec 4 | Paul Wedrich (ANU)

**On categorification of skein modules and algebras**

Khovanov homology and its cousins are usually defined as functorial invariants of links in \(\mathbb{R}^3\). Embracing their reliance on link projections as a virtue, they admit extensions to links in thickened surfaces, and, thus, categorify surface skein modules and, conjecturally, their algebra structures. Skein algebras are related to character varieties and quantum Teichmüller theory, and are the subject of positivity conjectures that appear in reach of categorification techniques. The focus of this talk will be recent joint work with Hoel Queffelec on functorial \(gl(2)\) surface link homologies.

November 27 | Louis Kaufmann (UIC)

**Unitary Braiding and Majorana Fermions**

Majorana fermions, particles that are their own anti-particles, were proposed by Ettore Majorana in 1937. Recently Majorana particles have become of interest in relation to the structure of anyons, topological quantum computing and the fundamental structure of standard Fermions such as electrons. The mathematics of this relationship associates to a standard Fermion, annihilation and creation operators \(\psi^*\) and \(\psi\) such that \(\psi^{*2} = \psi^2= 0\) and \(\psi \psi^* + \psi^* \psi = 1\). Behind this Fermion algebra are the Majorana operators \(a\) and \(b\) satisfying \(a^2 = b^2 = 1\) and \(ab + ba = 0\) making a Clifford algebra and \(\psi = (a + ib)/\sqrt{2} , \psi^* = (a-ib)/\sqrt{2}\). It has recently come to pass by experiments with electrons in nano-wires that the conjecture that a and b correspond to physical Majorana fermions underlying the electron is not without content. And one can consider the braiding of these Majorana particles. This talk will discuss the braid group representations that so arise and their relevance to quantum computing.

November 20 | Mikhail Khovanov (Columbia)

**How to categorify the ring of integers localized at two**

We'll describe a construction of a monoidal triangulated category with the Grothendieck ring isomorphic to \(\mathbb{Z}[1/2]\). This is a joint work with Yin Tian.

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.

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.