Spring 2022

- Date:
**2/4/22****No meeting due to faculty meeting**

- Date:
**2/11/22****Adrian Ioana, UCSD**- Title:
**Almost commuting matrices and stability for product groups** - Abstract: I will present a result showing that the direct product group $G=\mathbb F_2\times\mathbb F_2$, where $\mathbb F_2$ is the free group on two generators, is not Hilbert-Schmidt stable. This means that G admits a sequence of asymptotic homomorphisms (with respect to the normalized Hilbert-Schmidt norm) which are not perturbations of genuine homomorphisms. While this result concerns unitary matrices, its proof relies on techniques and ideas from the theory of von Neumann algebras. I will also explain how this result can be used to settle in the negative a natural version of an old question of Rosenthal concerning almost commuting matrices. More precisely, we derive the existence of contraction matrices A,B such that A almost commutes with B and B* (in the normalized Hilbert-Schmidt norm), but there are no matrices A’,B’ close to A,B such that A’ commutes with B’ and B’*.

- Date:
**2/18/22****Jesse Peterson, Vanderbilt University**- Title:
**Properly proximal von Neumann Algebras** - Abstract: Properly proximal groups were introduced recently by Boutonnet, Ioana, and the speaker, where they generalized several rigidity results to the setting of higher-rank groups. In this talk, I will describe how the notion of proper proximality fits naturally in the realm 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 Changying Ding and Srivatsav Kunnawalkam Elayavalli.

- Date:
**2/25/22****Hans Wenzl, UCSD**- Title:
**Categories for Quantum Group Fusion Categories** - Abstract: We present a method to construct large classes of module categories from deformations of embeddings of Lie groups $Sp(N)\subset SU(N)$ (N even) and $Sp(N-1)\subset SU(N)$ (N odd). Moreover, we can give detailed descriptions of these module categories. In particular, if the categories are unitarizable, we obtain finite depth subfactors whose first principal graphs are trunctions of the induction-restriction graphs of representations for the embeddings listed above.

- Date:
**3/4/22****Emily Peters, Loyola University Chicago**- Title:
**Conway's Rational Tangles and the Thompson group** - Abstract:
In the process of studying Thompson's group F (of piecewise
linear homeomorphisms from the closed unit interval [0,1] to itself,
which are differentiable except at finitely many dyadic rational
numbers), Vaughan Jones observed a map from F to knots. He proved
that every knot is in the image of this map -- that is, that every
knot can be seen as the "knot closure" of a Thompson group element.
Jones' algorithm to achieve this is rather piecemeal, and he asked if
there was a better one.

In a project with undergraduate student Ariana Grymski, we approach this question through the lens of Conway's rational tangles. We are able to give methods to construct any product or concatenation of simple tangles, and we hope these are seeds for a more skein-theoretic approach to the construction question.

- Date:
**3/11/22** **No Meeting, Spring Break**- Date:
**3/18/22****Dima Shlyakhtenko, UCLA**- Title:
**An Inequality for Non-Microstates Free Entropy Dimension for Crossed Products by Finite Abelian Groups** - Abstract: For certain generating sets of the subfactor inclusions coming from a crossed product by a finite abelian group we prove an approximate inequality between their non-microstates free entropy dimension, resembling the Shreier formula for ranks of finite index subgroups of free groups. As an application, we give bounds on free entropy dimension of generating sets of crossed products of a large class of von Neumann algebras by the direct sum of an infinite number of copies of the two-element group.

- Date:
**3/25/22****Special time: 3:00-5:00pm, on Zoom (same Zoom link).****Srivatsav Kunnawalkam Elayavalli, Vanderbilt University**- Title:
**Free entropy theory and rigid von Neumann algebras** - Abstract: Thesis defense.

- Date:
**4/1/22****Matthew Kennedy, University of Waterloo**- Title:
**The ideal intersection property for essential groupoid C*-algebras** - Abstract:
I will discuss recent work characterizing the ideal intersection property
for essential C*-algebras of étale groupoids with locally compact Hausdorff
space of units. For Hausdorff groupoids, this C*-algebra coincides with the
reduced C*-algebra. In the minimal case, the ideal intersection property is
equivalent to simplicity, so as a consequence we obtain a characterization
of étale groupoids that are "C*-simple.”

This is joint work with Se-Jin (Sam) Kim, Xin Li, Sven Raum and Dan Ursu.

- Date:
**4/8/22****This talk will be in SC 1431 (in person)!****Hui Tan, UCSD**- Title:
**Spectral gap characterizations of property (T) for II**_{1}factors - Abstract:
For property (T) II
_{1}factors, any inclusion into a tracial von Neumann algebra has spectral gap, and therefore weak spectral gap. I will discuss characterizations of property (T) for II_{1}factors by weak spectral gap in inclusions. I will explain how this is related to the non-weakly-mixing property of the bimodules containing almost central vectors, from which we also obtain a characterization of property (T).

- Date:
**4/15/22****Quan Chen, The Ohio State University**- Title:
**Q-system realization and applications** - Abstract: Q-systems are unitary versions of Frobenius algebra objects which appeared in the theory of subfactors. I will discuss higher unitary idempotent completion for C*/W* 2-categories called Q-system completion and its inverse 2-functor called realization. We will explain several applications of Q-system realization, including the coned construction to describe the inductive limit II_1 factor of a Jones tower and induced actions of unitary tensor categories on C*-algebras. Time permitting, we will explain the relationship between the equivariant K-theory and the K-theory of the realization.

- Date:
**4/19/22****Math Colloquium by Aldroubi-Azhari Prize Winner Cain Edie-Michell, 4:10pm to 5:00pm in SC 1206.****Cain Edie-Michell, UCSD**- Title:
**Representations of Quantum Group Categories** - Abstract: Given any algebraic object, it is important to study the representations of that object. This is particularly true for the tensor categories constructed from quantum groups. In this setting the representations classify certain conformal field theories, and give rise to highly non-trivial subfactors of many Von Neumann algebras. In this talk I will present some progress in the classification of these representations. Our results show the existence of several infinite families, along with a handful of sporadic examples.

- Date:
**4/22/22****Peter Huston, The Ohio State University**- Title:
**Composing topological domain walls and anyon mobility** - Abstract: Topologically ordered phases of matter are a subject which can be approached from a number of perspectives, including the perspective of lattice models, where topological order arises from systems of local operators on a Hilbert space. I will introduce lattice models of 2+1D topological order and 1+1D topological boundaries between them, and outline the relationship between these models and tensor categories which classify their topological order. A pair of parallel topological boundaries can decompose as a direct sum, a phenomenon which can be seen concretely in lattice models. In recent join work with Fiona Burnell, Corey Jones, and Dave Penneys, we have used the graphical calculus of 3-categories of enriched fusion categories to understand how this decomposition can be computed algebraically.

- Date:
**4/29/22****This talk will be in SC 1431 (in person)!****Ben Hayes, University of Virginia**- Title:
**Co-spectral radius, equivalence relations and the growth of unimodular random rooted trees** - Abstract: I will speak on joint work with Mikolas Abert and Mikolaj Fraczyk. In it, we define the co-spectral radius of inclusions $\cS\leq \cR$ of discrete, probability measure-preserving equivalence relations, using the sampling exponent of a generating random walk on the ambient relation. The co-spectral radius is analogous to the spectral radius for random walks on $G/H$ for inclusion $H\leq G$ of groups. Unlike the group case the almost sure existence of this pointwise limit is nontrivial to establish, and we provide a novel general method for proving almost sure existence of such limits based on the mass-transport principle. I will mention some analogies and connections with operator algebras and subfactor theory, as well as some applications.

- End of Spring Semester.