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:
**Essential Freeness of Stationary Actions of Lattices** - Abstract:
Let \Gamma be an irreducible lattice in a semisimple real Lie group G (with trivial center, at least two factors, at least one higher-rank). Stuck and Zimmer proved that every pmp action of such a lattice on a nonatomic space is essentially free; Peterson, building on joint work with myself on the p-adic case, generalized this to the non-commutative setting: every unitary representation of such a lattice on a II_1 factor is either finite-dimensional or is the left regular representation.
However, from the dynamical perspective, knowing that every measure-preserving action is free is not a complete answer: since lattices are nonamenable they have many natural actions which do not admit invariant measures. The natural question to ask is whether or not every minimal action of \Gamma on a compact metric space X admits a nonsingular measure so that the action is essentially free.

I answer this in the affirmative. Let \mu be the measure on \Gamma so that the Poisson boundary of (\Gamma,\mu) is that of G. We show that every action of \Gamma on a nonatomic probability space (X,\nu) where \nu is \mu-stationary is essentially free (such measures exist on every compact metric space where \Gamma acts).

I will also present some conjectures about the non-commutative generalization of this statement involving an apparently new notion of a ``stationary representation" of a lattice on a II_1 factor or Hilbert space that suggests that the rigidity phenomena of lattices is even stronger than is currently known.

- Date:
**4/13/18** **Marcel Bischoff, Ohio University**- Title:
**Infinite Index Subfactors from Conformal Inclusions** - Abstract: Conformal nets axiomatize chiral conformal field theory. A conformal inclusion is an inclusion of conformal nets, such that the local subfactors are irreducible. For example, for every conformal net A, there is a conformal inclusion of the Virasoro net of A inside A. The structure of finite index conformal inclusions is well understood. I will talk about some properties of infinite index arising from conformal inclusions and some open problems.
- 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:
**Tensor product decompositions of II**_{1}factors arising from amalgamated free product groups. - Abstract:
We show that a large collection of icc groups G give rise to II
_{1}factors which satisfy the following product rigidity phenomenon: all tensor product decompositions of the group von Neumann algebra L(G) must arise from canonical direct product decomposition of the underlying group G. This collection of groups include many remarkable groups studied throughout mathematics such as graph product groups, poly-amalgam groups, Burger-Mozes groups, Higman group, various integral two-dimensional Cremona groups, etc. As a consequence, we obtain several new examples of groups that give rise to prime factors. - End of Spring Semester.