In Alain Connes' noncommutative geometry, the normed ideals of compact operators are viewed as infinitesimals. A numerical invariant plays a key role in the study of operators modulo perturbations from these ideals. Recently new structure has appeared in these questions from the operator algebras which are commutants of selfadjoint n-tuples of operators modulo the normed ideal. Connections with dynamical entropy, K-theory of operator algebras, commutators of operators, supramenable groups and Banach space duality aspects will also be discussed. [Slides]

The free multiplicative Brownian motion $g_t$ is a non-normal process determined by the free SDE $dg_t = g_t\,dz_t$, where $z_t$ is a circular Brownian motion. It was introduced by Biane in 2001; he conjectured it was the large-$N$ limit (in $\ast$-distribution) of the Brownian motion on $\mathrm{GL}(N)$, and I proved this in 2016.
The big question is: what is the large-$N$ limit of the empirical eigenvalue distribution of this Brownian motion? Simulations show it is quite complicated, supported on a domain that is not simply connected for $t>4$.
The limit is (very likely) the Brown measure of $g_t$, which is a fierce object to compute. While the $\ast$-distribution determines the Brown measure, the Brown measure does not depend continuously on the $\ast$-moments, which makes it exceedingly difficult to compute or estimate.
Very recently, in joint work with Brian Hall, I have made substantial progress on computing (the support of) this Brown measure: we finally know what those tantalizing eigenvalue plots mean and where they come from. I will report in this research in progress. [Slides]

I will state the version of Voiculescu's noncommutative Weyl-von Neumann theorem for operators on $l^p$ I recently obtained. I will also give some consequences of this result.

In this talk a new notion of independence, which I call twisted independence, is introduced as a universal calculation rule for mixed moments of noncommutative random variables in the (modified) sense of Speicher. It is a complex valued one-parameter deformation (with modulus one) of the classical notion of independence. I will discuss some properties of twisted independence.

We adapt the theory of chordal Loewner chains to operator-valued matricial upper half-plane over a $C^*$ algebra. We show that operator-valued chordal Loewner chains are precisely the families of reciprocal Cauchy transforms that arise from processes with operator-valued monotone independent increments. We show that the Loewner equation \[ \partial_t F_{\mu_t}(z) = D_zF_{\mu_t}(z)[-G_{\nu_t}(z)], \] defines a bijection between (Lipschitz) Loewner chains $(F_{\mu_t})_{t \in [0,T]}$ and Herglotz vector fields $(-G_{\nu_t}(z))$; here $F_{\mu_t}$ is the reciprocal Cauchy transform of the operator-valued law $\mu_t$ and $G_{\nu_t}$ is the Cauchy transform of the operator-valued generalized law $\nu_t$. We give a combinatorial formula for the moments of the law $\mu_t$ in terms of the moments of the generalized law $\nu_t$. We realize the laws $\mu_t$ by operators on a modified monotone Fock space. We prove a version of the monotone central limit theorem which describes the behavior of $F_{\mu_t}$ for large $t$. [Slides]

Given an n-tuple of non-commutative random variables, its free Stein discrepancy relative to the semicircle law measures how "close" the distribution is to the semicircle law. By considering free Stein discrepancies relative to a broader class of laws, one can define a quantity called the free Stein information. In this talk, we will discuss this and its relation to other free probabilistic quantities such as the free Fisher information and the non-microstates free entropy dimension. This is based on joint work in progress with Ian Charlesworth.

I will present an analogue of Wigner’s moment method for invariant ensembles. In order to illustrate the method, I will use it to derive the Semicircle Law, the Marchenko-Pastur Law, and the limit shape law for random lozenge tilings. This is joint work with Sho Matsumoto. [Slides]

A few years ago in a landmark paper, Guionnet and Shlyakhtenko proved the existence of free monotone transport maps from the free group factors to von Neumann algebras generated by elements which have a joint law "close" to that of the free semicircular law. In this talk, I will discuss how to modify their idea to obtain similar results for interpolated free group factors using an operator-valued framework. This is joint work with Brent Nelson. [Slides]

I’ll present a few inequalities on metric spaces holding for $L_p$ and other natural spaces. Some of these inequalities can serve as the metric analogue of (Pisier’s) property $\alpha$ and used as an obstruction to the Lipschitz (and uniform) embeddability of (some discrete subsets of) Schatten classes into $L_p$ spaces. Joint work with Assaf Naor. [Slides]

In a joint work with C. Houdayer and S. Vaes, we give a complete classification of a large class of free Araki-Woods factors.

A subset $\Lambda$ of a discrete group $G$ is called "completely Sidon" (or "operator Sidon") if any bounded function $f:\Lambda\to B(H)$ extends to a c.b. map $\tilde f: C^*(G)\to B(H)$. Equivalently, the closed span of $\Lambda$ in $C^*(G)$, denoted by $C_\Lambda$, is completely isomorphic to the operator space version of the space $\ell_1$ (i.e. $\ell_1$ equipped with its maximal operator space structure). The typical example is a free set. Only non-amenable groups can contain infinite completely Sidon sets. Such sets have been previously considered by Bożejko. We generalize to this context Drury's classical theorem: completely Sidon sets are stable under finite unions. We also obtain the operator valued analogue of the "Fatou-Zygmund property": any bounded $f: \Lambda\to B(H)$ on an asymmetric completely Sidon set extends to a (completely) positive definite function on $G$. We give a completely isomorphic characterization of completely Sidon sets: $\Lambda $ is completely Sidon iff the operator space $C_\Lambda$ is completely isomorphic (by an arbitrary isomorphism) to $\ell_1(\Lambda)$. This is the operator space version of a result of Varopoulos for classical Sidon sets. We will also discuss the systems of non-commutative random variables that are "dominated by free-Gaussians," in analogy with the classical subGaussian systems.

King Yen Restaurant located at 2995 College Avenue

Anderson's paving conjecture, now known to hold due to the resolution of the Kadison-Singer problem, asserts that every zero diagonal Hermitian matrix admits non-trivial pavings with dimension independent bounds. In this paper show the existence of non-trivial simultaneous pavings for collections of k matrices with number of blocks depending polynomially on k, answering a question of Popa and Vaes. As a consequence, we get the correct asymptotic estimates for paving general zero diagonal matrices, and as an application, we give a simplified proof with slightly better estimates of a theorem of Johnson, Ozawa and Schechtman concerning commutator representations of zero trace matrices. Joint work with Mohan Ravichandran.

I will discuss some recent ongoing work with Paul Skoufranis to create a non-microstates bi-free entropy. I will propose a definition of bi-free conjugate variables and bi-free Fisher information, which admit desirable properties such as additivity in the presence of bi-free independence and versions of Cramer-Rao and Stam inequalities. I will also discuss the analogue of the free difference quotient, and some of the quirks present in the bi-free setting.

Recently a refinement of K-theory that provides invariants for vector bundles along with connections on them has been studied in geometry and topology. We will describe a version of this for the BDF theory Ext(X). Relations to questions in operator theory and generalized Toeplitz theory on contact manifolds will be discussed, as well as a possible extension to K-theoretic Spanier-Whitehead duality. This is part of a project with Ron Douglas and Xiang Tang and also one with Claude Schochet.