Spring 2021

- Date:
**1/29/21****Srivatsav Kunnawalkam Elayavalli**, Vanderbilt University- Title: On Heirs
- Abstract: We will introduce the notion of types and heirs in the context of embeddings into ultraproducts. As an application, we will prove that if any two embeddings of a Connes embeddable II_1 factor M into its own ultrapower are equivalent by an automorphism, then M is isomorphic to R. Time permitting, we will discuss the situation of embeddings into R^omega. Based on joint work with S. Atkinson and I. Goldbring.

- Date:
**2/5/21****Brent Nelson**, Michigan State University- Title: Free Stein dimension and von Neumann algebras
- Abstract: Let (M,\tau) be a tracial von Neumann algebra and A a finitely generated *-subalgebra of M. The free Stein dimension of (A,\tau) is the von Neumann dimension of a module determined by certain closable derivations on A. This quantity was originally defined as a free probabilistic invariant associated to the non-commutative distribution of generators for A with respect to \tau, but the perspective given by derivations on A is better suited for applications to von Neumann algebras such as detecting L^2-rigidity. In this talk, I will provide an introduction to free Stein dimension and show how its quantitative behavior can reveal structural properties of the von Neuman algebra A''. I will also discuss how to use the free Stein dimension to define a von Neumann algebra invariant, and present examples of when this invariant can be explicitly computed.

- Date:
**2/12/21****Corey Jones**, North Carolina State University- Title: Anomalous symmetries of geometric C*-algebras.
- Abstract: Given a group G and a U(1)-valued 3-cocycle w, an anomalous action on a C*-algebra is a generalization of a cocycle action of G, where the cocycle equations "hold up to w". We use the techniques of Eilenberg-MacLane, V. Jones, and Sutherland to show for every n>1 and every finite group G, every anomaly can be realized on the stabilization of a commutative C*-algebra of continuous functions on some closed connected n-manifold M. We also show that although there are no anomalous symmetries of Roe C*-algebras of coarse spaces, for every finite group G, every anomaly can be realized on the Roe corona of some bounded geometry metric space X with property A.

- Date:
**2/19/21****Julia Plavnik**, Indiana University- Title: Zesting link invariants
- Abstract: It was conjectured that modular categories were determined by its modular data (S- and T-matrices). In 2017, Mignard and Schauenburg presented a family of counterexamples to this conjecture, which led to the study of link invariants beyond modular data to distinguish these categories. In this talk we will discuss ribbon zesting, which is a construction of modular categories, and how it is related to the family of Mignard-Schauenburg counterexamples. To better understand this relation, we look into how zesting affects link invariants such as the W-matrix and the B-tensor. This talk is based on joint work in progress with Colleen Delaney and Sung Kim.

- Date:
**2/26/21**(11:10am - 12:30pm central)**Erik Christensen**, University of Copenhagen- Title: From spectral triples to rapid decay
- Abstract: A question on the domain of definition for a spectral triple has caused an interest in the Schur product of operator valued matrices. It then turned out that this is a completely bounded bilinear operator, which is a special case of the Hadamard product in a crossed product of a C*-algebra by a discrete group. This observation lead to an extension of the concept named rapid decay from group algebras to crossed products.

- Date:
**3/5/21****Ben Hayes**, University of Virginia- Title: A multiplicative ergodic theorem for von Neumann algebra valued cocycles
- Abstract: I'll discuss joint work with Lewis Bowen and Frank Lin. In it, we generalize the classical Multiplicative Ergodic Theorem (MET) of Oseledets to cocycles taking values in a semi-finite von Neumann algebra. In contrast with previous work, the limiting operator we get is typically not the exponential of a compact operator and often has diffuse spectrum.

- Date:
**3/12/21**(11:10am - 12:30pm central)**Sam Mellick**, ENS de Lyon- Title: Point processes on groups, their cost, and fixed price for G x Z.
- Abstract: Invariant point processes on groups are a rich class of probability measure preserving (pmp) actions. In fact, every essentially free pmp action of a nondiscrete locally compact second countable group is isomorphic to a point process. The cost of a point process is a numerical invariant that, informally speaking, measures how hard it is to "connect up" the point process. This notion has been very profitably studied for discrete groups, but little is known for nondiscrete groups.

This talk will not assume any sophisticated knowledge of probability theory. I will define point processes, their cost, and discuss why every point process on groups of the form G x Z has cost one. Joint work with Miklós Abért.

- Date:
**3/19/21****Sayan Das**, UC Riverside- Title: Property (T) factors with trivial fundamental group.
- Abstract: Motivated by Connes' rigidity conjecture, S. Popa conjectured in 2005 that fundamental groups of any property (T) group factor must be trivial. In this talk I shall give examples of Property (T) group factors with trivial fundamental group. This talk is partially based on a joint work with Ionut Chifan, Krishnendu Khan and Cyril Houdayer.

- Date:
**3/26/21**(11:10am - 12:30pm central)**Liviu Păunescu**, Institute of Mathematics of the Romanian Academy- Title: Recent results on P-stability
- Abstract: Two permutations that almost commute are close to two commuting permutations. The same question can be asked for other relations, not only the commutant. We shall see that the answer to this question depends only on the group that the equations describe. We then survey some recent results where this question is answered in positive or negative, depending on the group.

- Date:
**4/2/21****Rachel Norton**, Fitchburg State University- Title: Cartan subalgebras of twisted groupoid $C^*$-algebras and applications to higher rank graph $C^*$-algebras
- Abstract: In this talk we focus on Cartan subalgebras of groupoid $C^*$-algebras whose multiplication is twisted by a circle-valued $2$-cocycle. We identify sufficient conditions on a subgroupoid $S \subset G$ under which the twisted $C^*$-algebra of $S$ is a Cartan subalgebra of the twisted $C^*$-algebra of $G$. We then describe (in terms of $G$ and $S$) the so-called Weyl groupoid and twist that J. Renault defined in 2008, which give us a different groupoid model for our Cartan pair. We end with a discussion of our ongoing efforts to apply these results to higher rank graph $C^*$-algebras. This is joint work with A. Duwenig, E. Gillaspy, S. Reznikoff, and S. Wright.

- Date:
**4/9/21****Noah Snyder**, University of Indiana- Title: Connections in the language of Planar Algebras
- Abstract: The goal of this talk is to explain Ocneanu's language of biunitary connections and flat connections purely in Jones's language of planar algebras, and to give planar algebraic proofs of the main theorems about connections. In particular, I will introduce planar algebraic versions of module and bimodule categories, and show that a module is the same thing as a map to a graph planar algebra, while a bimodule corresponds to a biunitary connection. Finally, I'll show that a bimodule can be upgraded to a planar algebra if and only if the connection is flat. The main goal of this project is expository, but it also gives some new results because it lets us generalize the theory of connections to the non-unitary context using bi-invertible connections instead. Several of these ideas are anticipated in Planar Algebras I, but not stated in full generality due to issues around shadings. This is part of joint work with Dave Penneys and Emily Peters, in the paper we will give expositions in each of the tensor categorical language and planar algebra language, but since this is Subfactor seminar I will concentrate exclusively on the planar algebraic approach.

- Date:
**4/16/21****Alex Margolis**, Vanderbilt University- Title: Totally disconnected topological completions of quasi-actions
- Abstract: A well-known problem in group theory, dating back to work of
Furstenberg and Mostow, is to determine which locally compact groups
contain a fixed finitely generated group as a uniform lattice. We will
consider a geometric generalisation of this problem: classifying
cobounded quasi-actions on a fixed finitely generated group.

We introduce the concept of the topological completion of a quasi-action. This is a locally compact group, well-defined up to a compact normal subgroup, reflecting the geometry of the quasi-action. We exhibit a dichotomy in the class of hyperbolic groups, where either topological completions are always connected (when the group is a lattice in a Lie group) or are totally disconnected. We give applications to understanding the large scale geometry of finitely generated groups.

- Date:
**4/23/21****Cain Edie-Michell**, Vanderbilt University- Title: Type II quantum subgroups for sl_n
- Abstract: Quantum subgroups are module categories, which encode the ``higher representation theory'' of the Lie algebras. They appear naturally in mathematical physics, where they correspond to extensions of the Wess-Zumino-Witten models. The classification of these quantum subgroups has been a long-standing open problem. The main issue at hand being the possible existence of exceptional examples. Despite considerable attention from both physicists and mathematicians, full results are only known for sl_2 and sl_3.

In this talk I will discuss recent progress in the classification of type II quantum subgroups for sl_n. Our results finish off the classification for n = 4,5,6,7, and pave the way for higher ranks. In particular we discover several exceptional examples.

- Date:
**4/30/21****David Penneys**, The Ohio State University- Title: Classifying small index subfactors
- Abstract: Building on the work of many people over several decades, subfactor standard invariants have been classified up to index 5+1/4. We will give an overview of the history and techniques for subfactor standard invariant classification and construction.

- Date:
**5/7/21****Jun Yang**, Vanderbilt University- Title: The von Neumann Algebras from Cusp Forms Acting on Holomorphic Discrete Series
- Abstract: Given a Fuchsian group, its cusp forms intertwine its actions on the holomorphic discrete series representation of SL(2,R). We show these cusp forms can generate the entire commutant of its groups von Neumann algebra. We generalize this result to a semi-simple Lie group G with holomorphic discrete series and their lattices. We prove that the commutant is the tensor of its subalgebra from cusp forms and the endomorphism algebra of a highest weight representation V of a maximal compact subgroup of G. If the lattice is ICC, this gives a subfactor of type II_1 with index equals to of dim(V)^2.

- Date:
**5/14/21****Thomas Sinclair**, Purdue University- TItle: Lipschitz geometry of operator spaces
- Abstract: We will discuss a notion of Lipschitz embeddability of operator spaces which is strictly weaker than the usual linear notion of embeddability, yet is strong enough to capture aspects of the linear theory in many circumstances. This is joint work with Bruno Braga and and Javier Alejandro Chavez-Dominguez.

- Date:
**5/21/21**(11:10am - 12:30pm central)**Jan Spakula**, University of Southampton- Title: (Measured) asymptotic expanders and rigidity of Roe algebras
- Abstract: Let X be a countable discrete metric space, and think of operators on l^2(X) in terms of their X-by-X matrix. Band operators are ones whose matrix is supported on a "band" along the main diagonal; all norm-limits of these form a C*-algebra, called uniform Roe algebra of X. This algebra "encodes" the large-scale (a.k.a. coarse) structure of X. "Rigidity of Roe algebras" refers to the question whether isomorphism of Roe algebras implies coarse equivalence of the underlying spaces.

After a brief introduction of coarse geometry and Roe algebras, I will discuss quasi-locality, and how it leads to the notion of (measured) asymptotic expanders. This allows to (somewhat) generalise existing results about rigidity of Roe algebras, and provide (another) geometric rigidity criterion. If time permits, I will expand on some of the ingredients and techniques that we use. (Based on joint work with K. Li and J. Zhang.)

- Date:
**5/28/21****Matthew Zaremsky**, University of Albany (SUNY)- Title: Group von Neumann algebras of Thompson-like groups
- Abstract: I will discuss recent joint work with my grad student Eli Bashwinger, in which we analyze group von Neumann algebras arising from certain members of the extended family of Thompson's groups. Jolissaint proved that for Thompson's group F, the group von Neumann algebra L(F) is a McDuff type II_1 factor, and hence F is inner amenable. We use a construction called "cloning systems" to produce a large family of other Thompson-like groups with this same property. In particular we get the surprising result that the pure braided Thompson group bF is inner amenable. More generally, we produce a "machine" that takes as input an arbitrary countable group and outputs a Thompson-like group yielding a McDuff type II_1 factor.

- End of Spring Semester.