University Quantum Symmetries Lectures

( UQSL )

This is an online seminar organized by Corey Jones, David Penneys, and Julia Plavnik. Topics of interest range widely over "quantum" mathematics, and include (but are certainly not limited to):

  • Tensor categories

  • Subfactors and operator algebras

  • Hopf algebras and quantum groups

  • Representation theory

  • Higher Categories

  • TQFT and low dimensional topology

  • Categorification

  • Topological phases of matter

  • Conformal field theory

  • Algebraic quantum field theory

 

We typically meet Thursdays from 3:00 pm - 4:00 pm (Eastern Time Zone). If you would like to attend, please email Corey Jones at cormjones88@gmail.com to be added to the Google Group for weekly announcements of abstracts and Zoom links.

Here's a link to our schedule from previous semesters.

Below is a list of currently scheduled talks for this semester (which is updated frequently!):

Spring 2021:

1/14/2021- Michael Brannan, Texas A&M University:

Quantum graphs and quantum Cuntz-Krieger algebras.

I will give a light introduction to the theory of quantum graphs and some related operator algebraic constructions. Quantum graphs are generalizations of directed graphs within the framework of non-commutative geometry, and they arise naturally in a surprising variety of areas including quantum information theory, representation theory, and in the theory of non-local games. I will review the well-known construction of Cuntz-Krieger C*-algebras from ordinary graphs and explain how one can generalize this construction to the setting of quantum graphs.  Time permitting, I will also explain how quantum symmetries of quantum graphs can be used to shed some light on the structure of quantum Cuntz-Krieger algebras.  (This is joint work with Kari Eifler, Christian Voigt, and Moritz Weber.)  

1/21/2021- No talk this week (rescheduled for 3/4/2021).

1/28/2021- Victor Ostrik, University of Oregon:

Two dimensional topological field theories and partial fractions.

This talk is based on joint work with M.Khovanov and Y.Kononov. By evaluating a topological field theory in dimension 2 on surfaces of genus 0,1,2 etc we get a sequence. We investigate which sequences occur in this way depending on the assumptions on the target category.

2/4/2021- Georgia Benkart, University of Wisconsin - Madison

Fusion rules for Hopf algebras.

The McKay matrix M_V records the result of tensoring the simple modules with a finite-dimensional module V.   In the case of finite groups, the eigenvectors for M_V are the columns of the character table, and the eigenvalues come from evaluating the character of V on conjugacy class representatives.  In this talk, we will explore what can be said about such eigenvectors when the McKay matrix is determined by modules for a finite-dimensional Hopf algebra.  The quantum group u_q(sl_2), where  q is a root of unity, provides an interesting example.

2/11/2021 (TALK AT 2PM US EASTERN)- Xie Chen, California Institute of Technology:

Foliation structure in fracton models.

 

Fracton models are characterized by an exponentially increasing ground state degeneracy and point excitations with limited motion. In this talk, I will focus on a prototypical 3D fracton model -- the X-cube model -- and discuss how its ground state degeneracy can be understood from a foliation structure in the model. In particular, we show that there are hidden 2D topological layers in the 3D bulk. To calculate the ground state degeneracy, we can remove the layers until a minimal structure is reached. The ground state degeneracy comes from the combination of the degeneracy of the foliation layers and that associated with the minimal structure. We discuss explicitly how this works for X-cube model with periodic boundary condition, open boundary condition, as well as in the presence of screw dislocation defects.

2/18/2021- Theo Johnson-Freyd, Dalhousie University/Perimeter Institute:

 

Condensations and Components.

The 1-categorical Schur's lemma, which says that a nonzero morphism between simple objects is an isomorphism, fails for semisimple n-categories when n≥2. Rather, when two simple objects are related by a nonzero morphism, they each arise as a condensation descendant of the other. Because of this, for many purposes the natural n-categorical version of "set of simple objects" is the set of components: the set of simples modulo condensation descent. I will explain this phenomenon and describe some conjectures, including conjectures about "higher categorical S-matrices" and, time permitting, about the image of the j-homomorphism in the homotopy groups of spheres.
 

2/25/2021- Noah Snyder, Indiana University, Bloomington: 

 

Demystifying subfactor techniques for constructing tensor categories

The primary goal of this talk is to explain two techniques used for constructing tensor categories that were developed among the subfactor community by Asaeda--Haagerup and Jones--Peters in purely tensor categorical terms.  The secondary goal is to explain how these techniques generalize using module categories.  This talk is based on a part of our Extended Haagerup paper with Grossman-Morrison-Penneys-Peters (which builds on work of De Commer-Yamashita) and on work in progress with Penneys-Peters.

3/4/2021- Andy Manion, University of Southern California:

Higher representations and cornered Floer homology.

I will discuss recent work with Raphael Rouquier, focusing on a higher tensor product operation for 2-representations of Khovanov's categorification of U(gl(1|1)^+), examples of such 2-representations that arise as strands algebras in bordered and cornered Heegaard Floer homology, and a tensor-product-based gluing formula for these 2-representations expanding on work of Douglas-Manolescu.

3/11/2021- No talk this week.

3/18/2021- Zhenghan Wang, University of California Santa Barbara: TBA

3/25/2021- Eric Rowell, Texas A&M University: TBA

4/15/2021- Claudia Scheimbauer, Technische Universität München: TBA

4/22/2021- Radmila Sazdanovic, North Carolina State University: TBA

4/29/2021- Emily Riehl, Johns Hopkins University: TBA