Spring 2024

- Date:
**1/26/24****Corey Jones, North Carolina State University**- Title: Actions of fusion categories on noncommutative tori.
- Abstract: Noncommutative tori are a family of C*-algebras that provide some of the most accessible examples of simple nuclear C*-algebras that are not AF. In this talk, we will explain how to construct actions of the Haagerup-Izumi fusion categories on certain noncommutative 2-tori, and explain a no-go theorem which implies that most finite depth Temperley-Lieb-Jones categories cannot act on a noncommutative torus of any rank. Based on joint work with David Evans.

- Date:
**2/2/24**(Zoom link: https://vanderbilt.zoom.us/j/97468202440 )**David Jekel, The Fields Institute**- Title: Automorphisms and optimal transport theory for non-commutative probability spaces.
- Abstract: Much recent work in random matrix theory has focused on adapting optimal transport theory to the non-commutative setting; this for instance led to isomorphism of q-Gaussian factors for small values of q as well as free complementation for MASA in the free group algebra given by certain perturbations of one of the semicircular generators. The challenge is that, unlike in classical probability theory, the non-commutative moments of a tuple $(X_1,\dots,X_m)$ are not enough information to determine $(X_1,\dots,X_m)$ up to approximate automorphism. Thus, the interpretation of optimal transport theory as characterizing the distance between automorphism orbits is missing for the setting of non-commutative laws. In the non-commutative setting, automorphism orbits (in a suitable elementary extension) correspond to model-theoretic \emph{types}. We therefore propose that types are the correct non-commutative analog for probability distributions, and we develop a Monge-Kantorovich duality theorem, which gives an alternative characterization of the distance between orbits in terms of certain convex functionals.

- Date:
**2/16/24****Kai Toyosawa, Vanderbilt University**- Title: Weak exactness and amalgamated free products
- Abstract: Weak exactness for von Neumann algebras was first introduced by Kirchberg in 1995 as an analogue of exactness in the setting of C* -algebras. In this talk, I will show that the amalgamated free product of weakly exact von Neumann algebras is again weakly exact. The proof involves a universal property of Toeplitz-Pimsner algebras and a locally convex topology on bimodules of von Neumann algebras, which is used to characterize weak exactness.

- Date:
**2/23/24****Srivatsav Kunnawalkam Elayavalli, UCSD**- Title: Sequential Gamma for von Neumann algebras
- Abstract: We will introduce a generalization of property Gamma for von Neumann algebras that helps us understand the structure of ultrapowers better. Joint work with Greg Patchell.

- Date:
**3/1/24****Eli Bashwinger, University of Albany**- Title: Von Neumann Algebras from Thompson-like Groups
- Abstract: In this talk, I will discuss some results concerning the von Neumann algebras of Thompson-like groups arising from $d$-ary cloning systems (for $d \ge 2$). This talk will essentially be a survey of some of the many results coming from three papers I have written so far on this topic, the first two of which I wrote with Matthew Zaremsky, who invented cloning systems with Stefan Witzel in 2018. Given a $d$-ary cloning system on a sequence of groups $(G_n)_{n \in \mathbb{N}}$, we can form a Thompson-like group denoted by $\mathscr{T}_d(G_*)$, and this group canonically contains $F_d$, the smallest of the Higman-Thompson groups. The group inclusion $F_d \le \mathscr{T}_d(G_*)$ translates to an inclusion of their group von Neumann algebras $L(F_d) \subseteq L(\mathscr{T}_d(G_*))$. We have several results concerning this inclusion as well as some results concerning the von Neumann algebra $L(\mathscr{T}_d(G_*))$ itself, some of which will have group-theoretic consequences for these Thompson-like groups and how $F_d$ sits inside them.

- Date:
**3/8/24****Patrick Hiatt, UCLA**- Title: On the Singular Abelian Rank of Ultraproduct II$_1$ Factors
- Abstract: I will present some recent joint work with Sorin Popa where we show that, under the continuum hypotheses $\mathfrak{c} =\aleph_1$, any ultraproduct II$_1$ factor $M = \prod_{\omega} M_n$ of separable finite factors $M_n$ contains more than $\mathfrak{c}$ many mutually disjoint singular MASAs, in other words the singular abelian rank of $M$, ${\rm r}(M)$, is larger than $\mathfrak{c}$. Moreover, if the strong continuum hypothesis $2^\mathfrak{c}=\aleph_2$ is assumed, then ${\rm r}(M) = 2^\mathfrak{c}$. More generally, these results hold true for any II$_1$ factor $M$ with unitary group of cardinality $\mathfrak{c}$ that satisfies the bicommutant condition $(A_0'\cap M)'\cap M=M$, for all $A_0\subset M$ separable abelian.

- Date:
**3/15/24****No Meeting, Spring Break**

- Date:
**3/22/24, Special time: 3:10-4:00. Special room: SC1312.****Darren Creutz, U.S. Naval Academy**- Title: Adelic structure of low complexity subshifts
- Abstract: A subshift is a closed shift-invariant subset of $\{ 0, 1 \}^{\mathbb{Z}}$ or more generally $\mathcal{A}^{\mathbb{Z}}$ for some finite set $\mathcal{A}$, the alphabet. Subshifts are an interesting class in their own right which also arise in a natural way from arbitrary probability preserving actions from the coding by a (reasonable) partition. I will present a somewhat surprising result, joint with R. Pavlov, that subshifts of very low complexity are all measurably isomorphic to rotations on one-dimensional adelic compact groups. Consequences of this include that Sarnak's Conjecture holds for all low complexity subshifts and the resolution of an open question of Ferenczi about the minimal complexity for weak mixing.

- Date:
**3/22/24****Zhiyuan Yang, Texas A&M University**- Title: Twisted Araki-Woods algebras: factoriality via conjugate variables
- Abstract: The q-deformed Gaussian von Neumann algebras or more generally the Yang-Baxter deformed Gaussian von Neumann algebras were introduced in 1994 by Marek Bozejko and Roland Speicher as a deformation of free Gaussian algebras (isomorphic to the free group factors). In 2023, da Silva and Lechner generalized this Yang-Baxter deformation construction to the nontracial cases and the resulting algebras are called the twisted Araki-Woods algebras. We explain how the existence of conjugate variables implies the factoriality of finitely generated twisted Araki-Woods algebras using the powerful results of Brent Nelson on nontracial finite free fisher information. This generalizes the corresponding results for q-Gaussians by A. Miyagawa and R. Speicher and for q-Araki-Woods by M. Kumar, A. Skalski and M. Wasilewski. Should time permit, we will also talk about certain sufficient conditions for those algebras to have the Akemann-Ostrand property.

- Date:
**4/12/24****Jennifer Pi, UC Irvine**- Title: An Absence of Quantifier Reduction for $\textrm{II}_1$ Factors, using Quantum Expanders
- Abstract: A basic question in model theory is whether a theory admits any kind of quantifier reduction. One form of quantifier reduction is called model completeness, and broadly refers to when arbitrary formulas can be "replaced" by existential formulas. Prior to the negative resolution of the Connes Embedding Problem (CEP), a result of Goldbring, Hart, and Sinclair showed that a positive solution to CEP would imply that there is no $\textrm{II}_1$ factor with a theory which is model-complete. In this talk, we discuss work on the question of quantifier reduction for general tracial von Neumann algebras. In particular, we prove a complete classification for which tracial von Neumann algebras admit complete elimination of quantifiers. Furthermore, we show that no II$_1$ factor (satisfying a weaker assumption than CEP) has a theory that is model complete by using Hastings' quantum expanders. This is joint work with Ilijas Farah and David Jekel.

- Date:
**4/19/24****Changying Ding, UCLA**- Title: On Cartan subalgebras of $II_1$ factors arising from Bernoulli actions of weakly amenable groups
- Abstract: A conjecture of Popa states that the $II_1$ factor arising from a Bernoulli action of a nonamenable group has a unique (group measure space) Cartan subalgebra, up to unitary conjugacy. In this talk, I will discuss this conjecture and show that it holds for weakly amenable groups with constant $1$ among algebraic actions. The proof involves the notion of properly proximal groups introduced by Boutonnet, Ioana, and Peterson.

- Date:
**5/24/24****Michael Montgomery, Dartmouth College**- Title: An algebra structure for reproducing kernel Hilbert spaces
- Abstract: In the talk we will develop a natural definition (in the categorical sense) of an algebra structure on a reproducing kernel Hilbert space. This definition is also equivalent to subconvolutivity of weight functions in examples from harmonic analysis. We will then show the category of reproducing kernel Hilbert algebras (RKHA) is closed under orthogonal sums, tensor products, pushouts, and pullbacks such that the spectrum is a functor compatible with these constructions. Furthermore, the image of the spectrum as a functor from RKHA to Top contains all compact subspaces of R^n, n>0.

- Date:
**7/5/24****Arnaud Brothier, University of New South Wales, Sydney**- Title: Forest-skein groups
- Abstract: Motivated by Vaughan Jones' reconstruction program of conformal field theories, we have defined a class of groups constructed with tree-diagrams named "forest-skein (FS) groups". These groups are interesting on their own, satisfying exceptional properties and having powerful extra-structures for studying them. They are isotropy subgroups of forest-skein categories: categories of planar diagrams mod out by certain skein relations (just like planar algebras are in Jones' subfactor framework). We will present explicit examples and explain how the theory of forest-skein groups is surprisingly well-behaved. In particular, we will present new infinite families of infinite simple and finitely presented groups. Some results are joint work with Ryan Seelig.

- End of Spring Semester.