Fall 2022

- Date:
**9/2/22****Sayan Das, UC Riverside**- Title: On the invariant subalgebra rigidity (ISR) property
- Abstract: A group factor L(G) is said to have the ISR property, if every G invariant von Neumann subalgebra of L(G) arises from a normal subgroup of G. In this talk I will show that every G-invariant subfactor arises from a normal subgroup. I will also provide examples of a large class of icc groups, including hyperbolic groups, and nonamenable groups with positive first L^2-Betti number (containing an infinite amenable subgroup) whose corresponding group factors satisfy the ISR property. I will also mention some applications towards the study of invariant subalgebras of group C*-algebras, and discuss a few open problems. This talk is based on a recent joint work with Prof. Ionut Chifan and Bin Sun.

- Date:
**9/16/22****Sayan Das, UC Riverside**- Title: Poisson boundaries of finite von Neumann algebras and Popa's Mean Value Property
- Abstract: The study of noncommutative Poisson boundaries for finite von Neumann algebras was initiated by Prof. Jesse Peterson and the speaker in a recent work. In this talk, I will describe the construction of noncommutative Poisson boundaries, and present a double ergodicity theorem. I will also show how the double ergodicity theorem can be used to prove that every II_1 factor satisfies Popa's Mean-Value property, thereby answering a question he posed in 2019. If time permits, I will present some new results on the structure of noncommutative Poisson boundaries. This talk is based on a joint work with Prof. Jesse Peterson.

- Date:
**9/23/22****Changying Ding, Vanderbilt University**- Title: A topology on bimodules and properly proximal von Neumann Algebras
- Abstract: Properly proximal groups were introduced recently by Boutonnet, Ioana, and Peterson, where they generalized several rigidity results to the setting of higher-rank groups. In this talk, I will describe a topology on bimodules of von Neumann algebras and how this helps us to fit the notion of proper proximality in the setting of von Neumann algebras. I will also describe several applications, including that the group von Neumann algebra of a non-amenable inner-amenable group cannot embed into a free group factor, which solves a problem of Popa. This is joint work with Srivatsav Kunnawalkam Elayavalli and Jesse Peterson.

- Date:
**9/30/22****Srivatsav Kunnawalkam Elayavalli, UCLA**- Title: Two full factors with non isomorphic ultrapowers.
- Abstract: I will show you how to construct a full factor such that it and L(F_2) have non isomorphic ultrapowers. The construction uses a combination of techniques from deformation rigidity and free entropy theory. We also provide the first example of a II_1 factor that is full such that its ultrapower is strongly 1-bounded. This is joint work with Adrian Ioana and Ionut Chifan.

- Date:
**10/14/22****No meeting, Fall break**

- Date:
**10/21/22****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.

- Date:
**10/28/22****Michael Davis, University of Iowa**- Title: Rigidity for von Neumann algebras of graph product groups
- Abstract: I will discuss my ongoing joint work with Ionut Chifan and Daniel Drimbe on various rigidity aspects of von Neumann algebras arising from graph product groups whose underlying graph is a certain cycle of cliques and whose vertex groups are wreath-like product property (T) groups. In particular, I will describe all symmetries of these von Neumann algebras by establishing formulas in the spirit of Genevois and Martin's results on automorphisms of graph product groups. In doing so, I will highlight the methods used from Popa's deformation/rigidity theory as well as new techniques pertaining to graph product algebras.

- Date:
**11/4/22****Mehrdad Kalantar, University of Houston**- Title: On ideal and trace structures of C*-algebras generated by covariant representations
- Abstract: The talk is concerned with the general problems of determining ideal and trace structures of C*-algebras associated to minimal actions of discrete groups G on locally compact spaces X. We extend applications of Furstenberg boundary theory to this setup, and use that to completely determine simplicity and existence/uniqueness of traces on C*-algebras generated by covariant representations arising from stabilizer subgroups. These results are consequences of a more fundamental uniqueness property of a class of morphisms in the category of C*-dynamical systems, called `noncommutative boundary maps?. We will give applications in concrete examples. This is joint work with Eduardo Scarparo.

- Date:
**11/11/22****Corey Jones, North Carolina State University**- Title: K-theoretic classification of inductive limit actions of fusion categories on AF- algebras.
- Abstract: Actions of fusion categories on C*-algebras generalize actions of finite groups. We will explain a new K-theoretic invariant for fusion category actions and show that for inductive limits of finite dimensional actions of fusion categories on unital AF-algebras, this is a complete invariant. This extends Elliott's theorem classifying AF algebras by the ordered K0 to the ``symmetry-enriched" setting. As a consequence, we obtain a classification result for inductive limit actions of finite groups on AF-algebras.

- Date:
**11/18/22****Changying Ding, Vanderbilt University**- Title: First $\ell^2 $-Betti numbers and proper proximality
- Abstract: Properly proximal groups were introduced by Boutonnet, Ioana, and Peterson, where they generalized several rigidity results to the setting of higher-rank groups. In this talk, I will show that exact groups with positive first $\ell^2$-Betti numbers are properly proximal. I will also describe an OE-superrgidity result of Bernoulli shifts of nonamenable non-properly proximal exact groups.

- Date:
**11/25/22****No meeting, Thanksgiving break**

- Date:
**12/2/22****Peter Huston, Vanderbilt University**- Title: Fracton models, defect networks, and enriched fusion categories
- Abstract: Fracton phases of matter are classes of physical models, related to topological phases of matter, in which quasiparticles exhibit subdimensional mobility. Recently, networks of defects between topological phases have been proposed as a way of understanding fracton order in (3+1)D. In this talk, I will introduce fracton order from the perspective of topological defect networks. I will then describe recent joint work with Fiona Burnell and Dave Penneys on understanding (2+1)D slices of such defect networks using 3-categories of enriched fusion categories.

- End of Fall Semester.