Spring 2023

- Date:
**1/20/23****Changying Ding, Vanderbilt University**- Title: Biexact von Neumann algebras
- Abstract: Biexact groups were introduced by Ozawa in his solidity paper, where he showed primness of their group von Neumann algebras. In this talk, I will describe how the notion of biexactness fits naturally in the setting of von Neumann algebras and show biexactness for groups is stable under W*-equivalence. This is a joint work with Jesse Peterson.

- Date:
**2/24/23****Christopher Schafhauser, University of Nebraska-Lincoln**- Title: Homotopy groups of automorphism groups of classifiable C*-algebras.
- Abstract: In the early '90s, Elliott conjectured that separable simple nuclear C*-algebras are classified up to isomorphism by their K-theory groups and traces, in analogy with the Connes-Haagerup classification of separably acting injective factors by their type and flow of weights. In the last few years, Elliott's classification program has been completed under the two additional hypotheses: Z-stability and the UCT. In the von Neumann algebraic setting, the classification of injective factors was followed by a delicate analysis of the symmetries of such factors. For example, Ocneanu proved every amenable group admits a outer action on the hyperfinite II_1 factor R, which is unique up to cocycle conjugacy, and Popa and Takesaki proved the automorphism group of R is contractable. In the spirit of the latter result, I will discuss recent joint work with Jamie Gabe on the homotopy type of automorphism groups of "classifiable" C*-algebras.

- Date:
**3/3/23****Pinhas Grossman, UNSW Sydney** Zoom talk (4:10-5:30pm central time). Meeting ID: 974 6820 2440
- Title: Subfactors, quadratic categories, and modular data
- Abstract: Most known examples of finite depth subfactors can be described by symmetries of groups or quantum groups (together with some associated constructions). The main class of exceptions are the quadratic subfactors. A quadratic category is a fusion category which contains a unique non-trivial orbit under the tensor product action of its group of invertible objects. Evans and Gannon found striking patterns in modular data associated to Drinfeld centers of certain quadratic categories, and made some remarkable conjectures about the existence and structure of infinite families of such categories. In this talk, we will discuss some generalizations of the Evans-Gannon conjectures, and work-in-progress on constructing associated fusion categories. This is joint work with Masaki Izumi.

- Date:
**3/10/23****Pieter Spaas, University of Copenhagen** Zoom talk (11:00am-12:20pm central time). Meeting ID: 974 6820 2440
- Title: On compact extensions of W*-dynamical systems
- Abstract: During this talk, we are interested in group actions on tracial von Neumann algebras. We start with providing some equivalent characterizations of compact extensions in this setting, and use them to generalize several classical results from ergodic theory. We will indicate some of the difficulties one encounters when trying to generalize classical Furstenberg-Zimmer theory, and then provide some positive results, as well as counterexamples obstructing a completely analogous generalization. This is based on joint work with Asgar Jamneshan.

- Date:
**3/17/23****No Meeting, Spring Break**

- Date:
**3/24/23****Luca Giorgetti, University of Rome Tor Vergata** Zoom talk (11:00am-12:20pm central time). Meeting ID: 974 6820 2440
- Title: Haploid algebras, Q-systems, and the Schellekens list
- Abstract: Q-systems realized in a tensor category of endomorphisms or bimodules of a factor provide an alternative description of finite index subfactors. More generally, one can talk of abstract Q-systems (special C*-Frobenius algebras), Frobenius algebras, or just algebras in a unitary (i.e. C*) tensor category. We provide two unitarizability results for such algebras and their isomorphism classes. Namely, we show that a haploid (i.e. connected) algebra in a unitary tensor category is equivalent to a Q-system if and only if it is rigid (i.e. Frobenius). The haploid condition cannot be dropped from the statement. We also show that isomorphic algebras in the previously mentioned correspondence give rise to unitarily isomorphic Q-systems. Lastly, we present some of our motivating applications to the theory of unitary VOA extensions, and a tensor categorical proof that all the holomorphic VOAs with central charge c=24 corresponding to entries 1-70 in the list of Schellekens are unitary VOAs. Joint work with S. Carpi, T. Gaudio, R. Hillier, https://arxiv.org/abs/2211.12790

- Date:
**3/31/23****Matthew Kennedy, University of Waterloo**- Title: Noncommutative Choquet theory and noncommutative majorization
- Abstract: I will give an overview of noncommutative Choquet theory and discuss several applications to operator algebras. I will introduce a notion of noncommutative majorization, which leads to a multivariate generalization of the Schur-Horn theorem for finite von Neumann algebras.

- Date:
**4/7/23****Aldo Garciaguinto, Michigan State University**- Title: Free Stein Dimension of Crossed Product by Finite Group
- Abstract: In this talk, we will discuss a free probabilistic quantity called free Stein dimension and compute it for a crossed product by a finite group. The free Stein dimension is the Murray-von Neumann dimension of a particular subspace of derivations. Charlesworth and Nelson defined this quantity in the hope of finding a von Neumann algebra invariant. While it is still not known to be a von Neumann algebra invariant, it is an invariant for finitely generated unital tracial *-algebras and algebraic methods have been more successful than analytic ones in studying it. Our result continues this trend, and reveals a formula for the free Stein dimension of a crossed product by a finite group that is reminiscent of the Schreier formula for a finite index subgroups of free groups.

- Date:
**4/14/23****David Kribs, University of Guelph**- Title: Quantum error correction and operator algebras
- Abstract:Quantum error correction is a central topic in quantum information science. Its
origins as an independent field of study go back more than a quarter century,
and it now arises in almost every part of the subject, including most
importantly in recent years as a key area of focus in the development of new
quantum technologies.

A little over two decades ago, I began working in quantum information after receiving (excellent) doctoral and postdoctoral training primarily in operator theory and operator algebras. My initial works with several (very bright) collaborators focussed on bringing an operator algebra perspective and tools to the subject of quantum error correction. This led to the discovery of what's called 'operator' and 'operator algebra' quantum error correction (OAQEC), and related notions such as 'subsystem codes'. Recently, interest in the OAQEC approach has been renewed through applications in quantum gravity related investigations, and a recognition of it as an appropriate error correction framework for hybrid classical-quantum information processing implementations.

In this talk, I'll give a (brief) introduction to quantum error correction and the OAQEC formulation and its basic results. Time permitting, I'll also discuss an interesting connection with quantum privacy and some very recent related work I've been doing with Xanadu Quantum Technologies in Toronto.

- End of Spring Semester.