Fall 2023

- Date:
**9/1/23****Quan Chen, Vanderbilt University**- Title: Gauging for categorical Connes Chi(M)
- Abstract: Gauging a unitary braided tensor category (UBTC) in mathematics is a 2-step process: a G-crossed extension of a UBTC and then G-equivariantization of the resulting category, which ends up with a new UBTC. In 2111.06378, we constructed a unitary braiding on the tensor category Chi(M) from a II_1 factor M, extending both Connes' ordinary Chi(M) and Jones' kappa invariant. In this paper, we show that if M has a finite group G action with G neither centrally trivial nor approximately inner, then the gauging of Chi(M) is equivalent to Chi(M\rtimes G). This is joint work with Corey Jones and David Penneys.

- Date:
**9/8/23****Bat-Od Battseren, Vanderbilt University**- Title: Von Neumann equivalence and group exactness
- Abstract: Group exactness is a weaker version of amenability. It is known that exactness is stable under both Measure equivalence (ME) and W*-equivalence (W*E). Von Neuman equivalence (vNE) is a relatively recent notion that generalizes both ME and W*E. In this talk, we will see that group exactness is also stable under vNE, giving a unified proof for ME and W*E cases.

- Date:
**9/15/23****Dan Margalit, Vanderbilt University**- Title: Homomorphisms of Braid groups
- Abstract: In the early 1980s Dyer-Grossman proved that every automorphism of the braid group is geometric, meaning that it is induced by a homeomorphism of a punctured disk. In this talk, we will discuss this result, as well as generalizations in several directions: where the domain is a subgroup of the braid group, where the target is a larger braid group, and where the target is a finite group. All of the results can be interpreted in terms of maps of spaces of polynomials. The talk represents joint work with Lei Chen and Kevin Kordek. We will do our best to make the talk accessible to all attendees.

- Date:
**9/22/23****Julio Caceres, Vanderbilt University**- Title: New hyperfinite subfactors subfactors with infinite depth
- Abstract: We will discuss a new approach to classifying infinite depth subfactors coming from commuting squares. We will also use this approach to construct a new irreducible hyperfinite infinite depth subfactor with index (5+sqrt(17))/2. Using Kawahigashi's characterization of finite-dimensional commuting squares we also construct infinite depth subfactors with indices (5+sqrt(17))/2, 3+sqrt(3), (5+sqrt(21))/2, 5 and 3+sqrt(5) arising from 1-parameter families of bi-unitary connections.

- Date:
**9/29/23**Zoom talk 4:10pm-5:30pm**Ignacio Vergara, Universidad de Santiago de Chile**- Title: Property (T) for uniformly bounded representations
- Abstract: Kazhdan's Property (T) is defined in terms of unitary representations of locally compact groups. I will discuss a strengthening of this property by considering uniformly bounded representations instead. By doing this, one obtains a family of properties indexed by the value of this uniform bound. This allows us to associate, to every locally compact group, a constant defined as the critical value at which the group stops satisfying this stronger version of Property (T). For countable groups, we will see that this constant is invariant under von Neumann equivalence. I will also discuss some interesting examples, including rank 1 Lie groups and random hyperbolic groups.

- Date:
**10/6/23****Srivatsav Kunnawalkam Elayavalli, UC San Diego**- Title: Structure of free group factors.
- Abstract: I will speak about joint work with Hayes and Jekel where we show various new structural properties of the (more generally interpolated) free group factors, that strikingly generalize most of the currently known results. Some of these include the resolution of the coarseness conjecture, a vastly general strong solidity theorem, structure of ultraproduct embeddings of subalgebras, among others. The proofs use the Hayes' random matrix approach of the Peterson Thom conjecture, combined with abstract 1-bounded entropy permanence properties.

- Date:
**10/13/23****Junhwi Lim, Vanderbilt University**- Title: An index for quantum cellular automata on fusion spin chains
- Abstract: The index for 1D quantum cellular automata (QCA) was introduced to measure the flow of the information by Gross, Nesme, Vogts, and Werner. Interpreting the index as the ratio of the Jones index for subfactors leads to a generalization of the index defined for QCA on more general abstract spin chains. These include fusion spin chains, which arise as the local operators invariant under a global (categorical/MPO) symmetry, and as the boundary operators on 2D topologically ordered spin systems. We introduce our generalization of index and show that it is a complete invariant for the group of QCA modulo finite depth circuits for the fusion spin chains built from the fusion category Fib. This talk is based on a joint work with Corey Jones.

- Date:
**10/20/23****No meeting, Fall break**

- Date:
**10/27/23**, Zoom talk 11:10am-12:30pm.**Pieter Naaijkens, Cardiff University**- Title: Topological entanglement entropy and all that
- Abstract: Entropy has played an important role in the understanding of quantum phases of matter. For example ``topological entanglement entropy'? (TEE) is seen as a signature of topological order in quantum ground states. On the other hand, due to work by Pimsner and Popa (and later Hiai for Type III factors), we know that the Jones index of a subfactor can be expressed in terms of (relative) entropies. In this talk I will give an overview of topological entanglement entropy and outline how in concrete models there is a direct relation between TEE and the Jones index of a certain subfactor.

- Date:
**11/3/23****Cesar Cuenca, The Ohio State University**- Title: Random matrices and random partitions at varying temperatures Abstract: I will discuss the global-scale behavior of ensembles of random matrix eigenvalues and random partitions which depend on the "inverse temperature" parameter beta. The goal is to convince the audience of the effectiveness of the moment method via Fourier-like transforms in characterizing the Law of Large Numbers and Central Limit Theorems in various settings. We focus on the regimes of high and low temperatures, that is, when the parameter beta converges to zero and infinity, respectively. Part of this talk is based on joint projects with F. Benaych-Georges -- V. Gorin, and M. Dolega -- A. Moll.

- Date:
**11/10/23****Sayan Das, Embry-Riddle Aeronautical University**- Title: Jointly bi-harmonic functions on groups and Peripheral Poisson boundaries
- Abstract: The study of asymptotic properties of a random walk on a countable, discrete group G (with respect to a symmetric, generating probability measure) relies on understanding a natural boundary of the random walk, called the Poisson boundary. The study of Poisson boundaries is intimately related with the study of bounded harmonic functions on groups. The startling "double ergodicity theorem" of Kaimanovich states that (separately) bi- harmonic functions on groups is constant- a feature that has led to the discovery of many rigidity properties of group representations and group actions. This led Kaimanovich to ask about the characterization of (jointly) bi-harmonic functions. In this talk I will completely characterize bi-harmonic functions on groups, thereby answering Kaimanovich's question, that he posed in 1992. I will also introduce the notion of "Peripheral Poisson boundaries" , which was first considered in a recent paper of Bhat, Talwar and Kar (2022). I will completely characterize the peripheral Poisson boundaries of groups, thereby answering a recent question of Bhat, Talwar and Kar.

- Date:
**11/24/23****No meeting, Thanksgiving break**

- End of Fall Semester.