Spring 2018

- Date:
**2/9/18** **Jonathan Campbell, Vanderbilt University**- Title:
**Field Theories and Homotopy Theory** - Abstract: Recently, Freed and Hopkins gave a homotopy-theoretic classification of Symmetry Protected Phases (SPTs). Subsequently, I did a number of computations confirming that their classification agrees with classifications given by physicists (which, it should be noted, are achieved by markedly different methods). In this talk I'll explain the Freed-Hopkins classification and my computations. This talk is essentially an advertisement for homotopy theory as both a tool to conceptualize field theories and to do concrete computations.
- Date:
**2/16/18** **Valeriano Aiello, Vanderbilt University**- Title:
**The oriented Thompson group and the Homfly polynomial** - Abstract: Recently, Vaughan Jones discovered an unexpected connection between the Thompson groups and knots. Among other things, he showed that any oriented link can be obtained as the "closure" of elements of the oriented Thompson group $\vec{F}$. By using this procedure we show that certain specializations of the Homfly polynomial give rise to functions of positive type on $\vec{F}$ (up to a renormalization). This talk is based on a joint work with Roberto Conti (Sapienza Università di Roma) and Vaughan Jones (Vanderbilt University).
- Date:
**2/23/18** **Ben Hayes, University of Virginia**- Title:
**Algebraic actions: A max-min principle for local weak* convergence** - Abstract: I will discuss the entropy theory of algebraic actions of sofic groups. An algebraic action of a group G is an action by continuous automorphism of a compact group X. We typically think of these actions as probability measure-preserving actions, giving X the Haar measure. It is an interesting problem to find conditions when you can guarantee that the topological entropy and the measure-theoretic entropy of these actions agree. I will discuss some methods to attack this problem, which involve ultraproduct analysis, particularly the use of the Loeb measure space.
- Date:
**3/9/18** **No Meeting, Spring Break**- Date:
**3/16/18** **Brent Nelson, UC Berkeley**- Title:
**Free transport for interpolated free group factors** - Abstract: A few years ago in a landmark paper, Guionnet and Shlyakhtenko proved the existence of free monotone transport from the joint law of a free semicircular family. In particular, these results imply that the von Neumann algebra (resp. C*-algebra) generated by a free semicircular family is isomorphic to the von Neumann algebra (resp. C*-algebra) generated by self-adjoint operators with a joint law "close" to the semicircle law in a certain sense. Notably, the von Neumann algebra generated by a free semicircular family is a free group factor. In this talk, I will discuss how to obtain corresponding results for the interpolated free group factors using an operator-valued framework. This is joint work with Michael Hartglass.
- Date:
**3/23/18** **Ryan Vitale, Indiana University**- Title:
**Planar Algebra Presentations for a Family of Tensor Categories** - Abstract: We give a family of planar algebra presentations for subcategories of representations of $\mathbb{Z}$ and $\mathbb{F}_p$. These are similar in flavor to examples of small index subfactors, but differs slightly, for example the categories are not semisimple. There are some connections to the fundamental theorems of invariant theory; using planar algebra techniques we can compute generators and relations for the ring of invariants corresponding to the representation assigned to a strand in the planar algebra.
- Date:
**3/30/18** **Lauren Ruth, UC Riverside**- Title:
**Two new settings for examples of von Neumann dimension** - Abstract: Let $G=PSL(2,\mathbb{R})$, let $\Gamma$ be a lattice in $G$, and let $\mathcal{H}$ be an irreducible unitary representation of $G$ with square-integrable matrix coefficients. A theorem in Goodman--de la Harpe--Jones (1989) states that the von Neumann dimension of $\mathcal{H}$ as a $W^*(\Gamma)$-module is equal to the formal dimension of the discrete series representation $\mathcal{H}$ times the covolume of $\Gamma$, calculated with respect to the same Haar measure. We will present two results inspired by this theorem. First, we show there is a representation of $W^*(\Gamma)$ on a subspace of cuspidal automorphic functions in $L^2(\Lambda \backslash G)$, where $\Lambda$ is any other lattice in $G$, and $W^*(\Gamma)$ acts on the right; and this representation is unitarily equivalent to one of the representations in [GHJ]. Next, we calculate von Neumann dimensions when $G$ is $PGL(2,F)$, for $F$ a local non-archimedean field of characteristic $0$ with residue field of order not divisible by 2; $\Gamma$ is a torsion-free lattice in $PGL(2,F)$, which, by a theorem of Ihara, is a free group; and $\mathcal{H}$ is the Steinberg representation, or a depth-zero supercuspidal representation, each yielding a different dimension.
- Date:
**4/6/18** **Darren Creutz, U.S. Naval Academy**- Title:
- Abstract:
- Date:
**4/13/18** **Sujan Pant, Alvernia University**- Title:
- Abstract:
- Date:
**April 14 & 15, 2018**,**AMS Southeastern Sectional Meeting at Vanderbilt University****Special Session***Advances in Operator Algebras*(organized by S. Atkinson, D. Bisch, V. Jones, J. Peterson)**Special Session***Function Spaces and Operator Theory*(organized by C. Chu, D. Zheng)**Special Session***Tensor Categories and Diagrammatical Methods*(organized by M. Bischoff, H. Tucker)

- Date:
**4/20/18** **Rolando de Santiago, UCLA**- Title:
- Abstract:
- End of Spring Semester.