Fall 2019

- Date:
**8/30/19****Cain Edie-Michell, Vanderbilt University**- Title: Classical Symmetries and Quantum Subgroups
- Abstract: In the early 2000's Ocneanu initiated the classification of quantum subgroups of a Lie algebra $\mathfrak{g}$ by providing a complete classification of quantum subgroups of the algebras $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$. Despite a large effort since then, little progress has been made for the other simple Lie algebras. However, recent results of Gannon and Schopieray have blown this problem wide open, by providing effective bounds on the levels at which quantum subgroups can appear for a given Lie algebra $\mathfrak{g}$. Hence there has been a recent revival in the program to classify all quantum subgroups, and in particular, to construct examples that are predicted to exist.

In this talk I will describe the connection between symmetries of the category of level $k$ integrable representations of $\mathfrak{g}$, and the quantum subgroups of $\mathfrak{g}$ appearing at level $k$. In particular I will give constructions of the conjectured*charge conjugation*quantum subgroups of $\mathfrak{sl}_n$ at all levels, and of a sporadic quantum subgroup of $\mathfrak{g}_2$ which appears at level 4.

- Date:
**9/13/19****Luca Giorgetti, Vanderbilt University**- Title: Minimal index and dimension for inclusions of von Neumann algebras with finite-dimensional centers
- Abstract: The notion of index for inclusions of von Neumann algebras goes back to the seminal work of Jones on subfactors of type II1. More generally, one can define the index of a conditional expectation associated with a subfactor and look for expectations that minimize the index. This minimal value is a number and it is called the minimal index of the subfactor.

We report on our analysis of the minimal index for inclusions of arbitrary von Neumann algebras (not necessarily factorial, nor finite) with finite-dimensional centers (multi-factor inclusions). The theory is controlled by a matrix associated with the inclusion, that we call matrix dimension, whose squared L^2 norm equals the minimal index and which determines a further invariant, that we call spherical state of the inclusion, via Perron-Frobenius theory. We mention the properties of finite multi-factor inclusions, especially multi-matrices, for which the spherical state coincides on the relative commutant with the Markov trace (super-extremal inclusions). We also mention how the matrix dimension can be purely algebraically defined for 1-arrows in rigid 2-C*-categories and how it determines the so-called standard solutions of the conjugate equations, and we address some open questions. Based on joint work with R. Longo, arXiv:1805.09234.

- Date:
**9/20/19****Srivatsav Kunnawalkam Elayavalli, Vanderbilt University**- Title: A plethora of embeddings into an ultraproduct of II_1 factors.
- Abstract: We will report on some results (joint with S. Atkinson) regarding new characterizations of amenability for Connes embeddable II_1 factors. First we introduce the notion of self-tracial stability and discuss immediate connections with amenability. Secondly we will discuss an interesting strengthening of the well known Jung's tubularity result. The core of this is a technical argument of Kishimoto. There are also some interesting model theoretic questions that come out of this. Finally we build on work of N. Brown and N. Ozawa to answer a question of Popa concerning the "vastness" of the space of embeddings of a non-amenable II_1 factor into the ultraproduct of a given collection of II_1 factors. We will end with interesting questions in the group theory setting. See arxiv: 1907.03359 for submitted paper.

- Date:
**9/27/19****Jun Yang, Vanderbilt University**- Title: Motzkin Algebra and the Fusion Rule A_n of Its Bimodules.
- Abstract: There is a family of algebras associated with the Motzkin numbers and a parameter d. We will prove that one can construct a hyperfinite $II_1$ factor from them if and only if d is $2cos(\pi/n)+1, (n\geq 3)$ or no less than 3. We will construct a sequence of bimodules over it. Then we will discuss the irreducibility, dimensions and get a fusion rule of $A_n$. This is joint work with Vaughan Jones.

- Date:
**10/4/19****Anton M. Zeitlin, Louisiana State University**- Title: Super-Teichmueller spaces and related structures.
- Abstract: The Teichmueller space parametrizes Riemann surfaces of fixed topological type and is fundamental in various contexts of mathematics and physics. It can be defined as a component of the moduli space of flat G=PSL(2,R) connections on the surface. Higher Teichmueller space extends these notions to appropriate higher rank classical Lie groups G, and N=1 super-Teichmueller space likewise studies the extension to the super Lie group G=OSp(1|2). In this talk, I will discuss the solution to the long-standing problem of giving Penner-type coordinates on super-Teichmueller space and its higher analogues and will also talk about several applications of this theory including the recent generalization of the McShane identity to the super case.

- Date:
**10/11/19** - Date:
**10/18/19** - Date:
**10/25/19****No meeting, Fall break**

- Date:
**11/1/19** - Date:
**11/8/19** - Date:
**11/15/19****Brent Nelson, Michigan State University**

- Date:
**11/22/19****Bin Gui, Rutgers University**

- Date:
**11/29/19****No meeting, Thanksgiving break**

- Date:
**12/6/19** - End of Fall Semester.

Some related conferences/workshops this semester: