## Program

__Tuesday, March 22__

## 09:00-09:30 | Coffee

## 09:30-10:10 | **Dan-Virgil Voiculescu:** __Free and Bi-Free Extremes__

The talk will be about extreme values in free probability and the beginning of an extension to bi-free probability, that is to free probability with left and right variables.

## 10:20-11:00 | **Jamie Mingo:** __Bi-free Infinite Divisibility__

We give a characterization of bi-free infinite divisibility for probability measures on $\mathbb{R}^2$. This is joint work with Yinzheng Gu and Hao-Wei Huang. Given a pair of commuting self-adjoint operators $a$ and $b$ in a unital C$^*$-algebra, their joint distribution with respect to a state $\varphi$ is described by a probability measure $\rho$ on $\mathbb{R}^2$. Such a pair of operators defines a pair of faces in the sense of Voiculescu's theory of bi-freeness, which will be bipartite. We can then ask when such a pair of faces is bi-freely infinitely divisible. This question can be answered in term of a conditionally positive definite condition on the bi-free cumulants of $\rho$, and a bi-free version of Bercovici's and Voiculescu's Levy-Khintchine equation. [Slides]

## 11:10-11:30 | Coffee Break

## 11:30-12:10 | **Paul Skoufranis:** __On Operator-Valued Bi-Free Distributions__

In this talk, distributions of bi-free pairs will be discussed in the operator-valued setting. In particular, the necessary joint moments and cumulants required to completely describe such distributions will be examined. Furthermore, if $D$ is a unital subalgebra of a unital algebra $B$, simple conditions on the bi-free cumulants may be described in order to guarantee a pair of operators $(X, Y)$ is bi-free from $(B, B^{\mathrm{op}})$ over $D$. This leads to further examples of bi-free pairs of matrices. Finally, the operator-valued bi-free partial transforms will be discussed. [Slides]

## 12:20-01:50 | Lunch

## 01:50-02:30 | **Hari Bercovici:** __Around the Connes Embedding Problem__

We consider a few conjectures related with the embedding problem. For instance, one way to approach embedding is to show that there exists a bijection between equivalence classes of pairs of selfadjoint variables in a W*-probability space and a class of concave functions called hives. The finite dimensional versions of these conjectures have not been proved, except in special cases which we may be able to describe. This is work in progress, joint with W. S. Li. [Slides]

## 02:40-03:20 | **Kevin Schnelli:** __Local law of addition of random matrices on optimal scale__

Describing the eigenvalue distribution of the sum of two general Hermitian matrices is basic question going back to Weyl. If the matrices have high dimensionality and are in general position in the sense that one of them is conjugated by a random Haar unitary matrix, the eigenvalue distribution of the sum is given by the free additive convolution of the respective spectral distributions. This result was obtained by Voiculescu on the macroscopic scale. In this talk, I show that it holds on the microscopic scale all the way down to the eigenvalue spacing. Joint work with Zhigang Bao and Laszlo Erdos. [Slides]

## 03:30-04:00 | Coffee Break

## 04:00-04:40 | **Emily Redelmeier:** __Connections between higher-order free cumulants and matrix cumulants__

Matrix cumulants, defined by Capitaine and Casalis in '06, may be interpreted as the contributions of surfaces in a topological expansion. This picture suggests a further subdivision in the matrix cumulants into vertex factors and interaction terms of various orders. In many important matrix models, interaction terms between higher numbers of vertices are less significant in terms of the dimension N of the matrix. I will discuss connections between these vertex interaction terms and higher-order free cumulants.

## 04:50-05:30 | **Claus Koestler:** __Unitary representations of the Thompson group $F$__

We present a new construction of unitary representations of the Thompson group $F$ and give results towards a classification of a large class of such representations. This is joint work with B. V. Rajarama Bhat.

__Wednesday, March 23__

## 09:00-09:30 | Coffee

## 09:30-10:10 | **Dimitri Shlyakhtenko:** __Free probability of type B and asymptotics of finite-rank perturbations of random matrices__

We show that finite rank perturbations of certain random matrices fit in the framework of infinitesimal (type B) asymptotic freeness. This can be used to explain the appearance of free harmonic analysis (such as subordination functions appearing in additive free convolution) in computations of outlier eigenvalues in spectra of such matrices. [Slides]

## 10:20-11:00 | **Benoit Collins:** __Free probability for random matrices with purely discrete eigenvalues__

In random matrix theory, some important theorems of free probability flavour can be stated as follows: a multi-matrix model that admits a joint limiting distribution in the limit of large dimension can be enlarged by adding appropriate random matrices (independent, unitarily invariant, etc) in such a way that the new enlarged family still converges in distribution, and that the limiting distribution of the enlarged family can be computed with an appropriate notion of (free) independence.

In this talk, we extend this paradigm to the different context of ‘pure spike' random matrices. Essentially, we are interested the case where the non-normalized trace of any non-commutative polynomial in the random variables converge (pure spike random matrices). We describe natural assumptions to enlarge such families of random matrices with the help ‘classical' random matrices, and obtain an almost sure pure spike convergence theorem for these enlarged families. We discuss methods to compute the enlarged ‘limiting spike distribution’, and their relation to free probability. Joint work with Takahiro Hasebe and Noriyoshi Sakuma, arXiv:1512.08975

## 11:10-11:30 | Coffee Break

## 11:30-12:10 | **J. William Helton:** __Noncommutative bianalytic maps on free convex sets__

The goal is to see if there are many or just a few free bianalytic maps of one free set to another. Our main concern is with free convex sets and we have a natural outline for classifying the bianalytic maps $f$ between them. It is a path requiring development of several areas: free convex sets are defined by affine Linear Matrix Inequalities (LMIs). Given two LMIs, say $L_A$ and $L_B$, a bianalytic map $f$ between the corresponding free convex sets $D_A$ and $D_B$ induces a defining function $p$ on $D_A$. There turns out to be a strong algebraic relationship between the two defining functions $p$ and $L_A$, called a Positivstellensatz. There is a statespace representation for key functions in the Positivstellensatz. Finally comes an analysis of this representations and the geometry of the boundary of $D_A$. The talk will emphasize one of the above areas. The work is with Igor Klep, Scott McCullough, Meric Augat.

## 12:20-01:50 | Lunch

## 01:50-02:30 | **Greg Anderson:** __Cumulants as Grammars__

Schutzenberger defined the notion of an algebraic noncommutative power series a half-century ago to generalize the notion of a context free language. Generalizing a question raised in a paper of Shlyakhtenko and Skoufranis, one may ask if algebraicity of the moment generating function of a family of noncommutative random variables is stable under operations typical in free probability. The answer is “yes,” as we will explain. Algebraicity is of interest because it has strong consequences for the regularity of a law. But more is true: “parse trees are decorated noncrossing partitions.” Time permitting we will explain some consequences of the latter intuition linking theoretical computer science to combinatorial free probability. (Joint work with Ohad Feldheim.)

## 02:40-03:20 | **Serban Belinschi:** __A noncommutative version of the Julia-Caratheodory Theorem__

The classical Julia-Caratheodory (or Julia-Wolff-Caratheodory) Theorem characterizes the behaviour of the derivative of an analytic self-map of a unit disc or of a half-plane of the complex plane at certain boundary points. In this talk, we shall provide a version of this result that applies to noncommutative self-maps of noncommutative half-planes of von Neumann algebras at points of the distinguished boundary of the domain. The proofs make use in an essential way of the noncommutative structure of the domains and maps involved. We shall conclude the talk with some examples inspired by free probability. [Slides]

## 03:30-04:00 | Coffee Break

## 04:00-04:40 | **Ian Charlesworth:** __Regularity of polynomials in non-commuting random variables__

Given an $n$-tuple of non-commuting random variables $y_1, \ldots, y_n$ and a non-constnat self-adjoint polynomial $P$ in $n$ indeterminates, we set $y = P(y_1, \ldots, y_n)$ and ask how the behaviour of $y$ is affected by properties of $y_1, \ldots, y_n$ and $P$. It turns out that if $y_1, \ldots, y_n$ are free, algebraic, and have finite free entropy, so too does $y$. If instead we assume that $y_1, \ldots, y_n$ have a dual system, then the spectral measure of $y$ has support which is not Lebesgue null, and if $P$ is homogeneous (e.g,. a monomial) then the spectral measure of $y$ is Lebesgue absolutely continuous. This is joint work with Dimitri Shlyakhtenko.

## 04:50-05:30 | **John Williams:** __Operator-Valued Free Probability for Unbounded Operators__

I will discuss the extension of operator valued free probability to operators affiliated with a tracial von Neumann algebra. In particular, the R-transorm will be defined and summability will be proven. More recent advances will include extending subordination to this setting.

## 06:30-08:30 | Conference Dinner

King Yen Restaurant located at 2995 College Avenue

__Thursday, March 24__

## 09:00-09:30 | Coffee

## 09:30-10:10 | **Steven N. Evans:** __The fundamental theorem of arithmetic for metric measure spaces__

A metric measure space (mms) is simply a complete, separable metric space equipped with a probability measure that has full support. A fundamental insight of Gromov is that the space of such objects is much "tamer" than the space of complete, separable metric spaces per se because mms carry within themselves a canonical family of approximations by finite structures: one takes the random mms that arises from picking some number of points independently at random and equipping it with the induced metric and uniform probability measure. A natural (commutative and associative) binary operation on the space of mms is defined by forming the Cartesian product of the two underlying sets equipped with the sum of the two metrics and the product of the two probability measures. There is a corresponding notion of a prime mms and an analogue of the fundamental theorem of arithmetic in the sense that any mms has a factorization into countably many prime mms which is unique up to the order of the factors. This is joint work with Ilya Molchanov (Bern). [Slides]

## 10:20-11:00 | **Ken Dykema:** __On algebra-valued R-diagonal elements__

R-diagonal elements in usual, scalar-valued *-noncommutative probability spaces include some of the non-self-adjoint operators that have been among those most studied in free probability theory. Analogous elements in the algebra-valued setting (also called operator-valued or amalgamated setting) were introduced by Sniady and Speicher in 2001, but have not been much investigated since then. A case of them arose naturally in studies of certain random Vandermonde matrices. In this talk, we describe characterizations of algebra-valued R-diagonal elements. One of these is in terms of non-crossing cumulants. We will also describe some results and examples concerning *-freeness of the unitary part and absolute values arising in the polar decompositions of some algebra-valued R-diagonal elements and algebra-valued circular elements. (Joint work with March Boedihardjo.)

## 11:10-11:20 | Coffee Break

## 11:20-12:00 | **March Boedihardjo:** __Asymptotic moments of random Vandermonde matrix__

We start by recalling the definition and some properties of the random Vandermonde matrix. We then describe their moments in terms of R-diagonality with amalgamation. Joint work with Ken Dykema. [Slides]

## 12:10-12:50 | **Todd Kemp:** __Invariant $\mathrm{GL}(N,\mathbb{C})$ Diffusions, the Segal--Bargmann Transform, and the Large-$N$ Limit__

For each $N\ge 2$, there is a two-parameter family of diffusion processes on $\mathrm{GL}(N,\mathbb{C})$ that is invariant under conjugation by $\mathrm{U}(N)$. The large-$N$ limits and fluctuations of these processes were studied by Guillaume C\'ebron and me, and partially presented at FPLNL4. The biggest remaining open question is to understand the large-$N$ limits of the empirical spectral measures of these processes.

In this talk, I will focus on the work of my current PhD student Ching Wei Ho, who has studied the free Segal--Bargmann transform associated to the large-$N$ limits of these diffusions. Using the now well-developed complex analytic methods of subordination, it is possible to characterize the range of the transform completely (at all times except $4$), and the corresponding domains in $\mathbb{C}$ are connected to the Brown measures of the free diffusions.

## 01:00-00:00 | Free Afternoon

__Friday, March 25__

## 09:00-09:30 | Coffee

## 09:30-10:10 | **Alexandru Nica:** __Eta-diagonal distributions, and infinite divisibility for R-diagonals__

We look at R-diagonal *-distributions, and we consider the concept of infinite divisibility for such a distribution, with respect to the operation of free additive convolution. We study how the method of the Boolean Bercovici-Pata (BBP) bijection applies to the study of this kind of infinite divisibility. To this end, it is natural to introduce a concept of "eta-diagonal distribution", which is the counterpart of R-diagonal distribution in the parallel (simpler) world of Boolean probability. We establish a number of properties of eta-diagonal distributions, then we follow on how BBP relates eta-diagonal distributions to infinitely divisible R-diagonal ones. This is joint work with Hari Bercovici, Michael Noyes and Kamil Szpojankowski.

## 10:20-11:00 | **Nikhil Srivastava:** __Concentration of Covariance Matrices for Distributions with $2+\varepsilon$ moments__

We study the minimal sample size $N=N(n)$ that suffices to estimate the covariance matrix of an $n$-dimensional distribution by the sample covariance matrix in the operator norm, with an arbitrary fixed accuracy. A result of Rudelson shows that $N=O(n\log{n})$ is necessary and sufficient in general. We show that under the stronger assumption that the marginals of the distribution have bounded $2+\varepsilon$ moments, a bound of $N=O(n)$ is possible. Interestingly, the lower edge only requires control on 1-dimensional marginals whereas the upper edge requires a stronger assumption on marginals of all dimensions. The proof is based on evaluating the Stieltjes transform at carefully chosen real random points outside the spectrum. Joint work with Roman Vershynin. [Slides]

## 11:10-11:30 | Coffee Break

## 11:30-12:10 | **Benjamin Hayes:** __1-Bounded Entropy and Regularity Problems in von Neumann Algebras__

We introduce and investigate the singular subspace of an inclusion of a tracial von Neumann algebra $N$ into another tracial von Neumann algebra $M$. The singular subspace is a canonical $N-N$ subbimodule of $L^2(M)$ containing the normalizer, the quasi-normalizer (introduced by Izumi-Longo-Popa), the one-sided quasi-normalizer (introudced by Fang-Gao-Smith), and the $wq$-normalizer (introduced by Galatan-Popa). By abstracting Voiculescu's original proof of absence of Cartan subalgebras, we show that the von Neumann algebra generated by the singular subspace of a diffuse, hyperfinite subalgebra is not $L(\mathbb{F}_2)$. We rely on the notion of being strongly 1-bounded, due to Jung, and the 1-bounded entropy, a quantity which measures "how" strongly 1-bounded an algebra is. Our methods are robust enough to repeat this process by transfinite induction and we use this to prove some conjectures made by Galatan-Popa in their study of smooth cohomology of $\rm{II}_1$-factors. We also present applications to nonisomoprhism problems for Free-Araki woods factors, as well as crossed products by Free Bogoliubov automorphisms in the spirit of Houdayer-Shlyakhtenko.

## 12:20-01:50 | Lunch

## 01:50-02:30 | **Brent Nelson:** __An Improved Non-Microstates Free Logarithmic Sobolev Inequality__

Using ideas from a 2014 paper of Ledoux, Nourdin, and Peccati, I will improve the free log-Sobolev inequality, originally established by Biane and Speicher in 2001, which pertains to the non-microstates free entropy of an $n$-tuple of non-commutative random variables in a tracial $W^*$-probability space $(M,\tau)$. The key idea is to consider the free analogue of a "Stein kernel," which arise classically in the context of Stein's method. A free Stein kernel is a $n\times n$ matrix with entries in the Hilbert–Schmidt operators on $L^2(M,\tau)$. These objects have appeared in every example of free transport thus far; namely, their proximity to the identity matrix with respect to a particular Banach norm is necessary for there to exist free transport to a free semicircular family. This is joint work with Max Fathi.

## 02:40-03:20 | **Qiang Zeng:** __Mixed $q$-Gaussian algebras and free transport__

The so-called mixed $q$-Gaussian algebras are the von Neumann algebras generated from the $q_{ij}$-commutation relation introduced by Speicher in 1993. In this talk, I will report some recent results for these algebras. For example, they have the complete metric approximation property and are strongly solid in the sense of Ozawa and Popa for $|q_{ij}|\lt 1$. They are in fact isomorphic to the free group factor $L(\mathbb{F}_N)$ provided $q_{ij}$'s are small in a certain sense. Here $N \in \{2,3,\ldots\}\cup \{+\infty\}$ is the number of generators. The infinite generator case is proved by extending the free transport theory of Guionnet and Shlyakhtenko to the case of infinite variables. Joint work with Marius Junge and Brent Nelson. [Slides]

## 03:30-04:00 | Coffee Break

## 04:00-04:40 | **Michael Hartglass:** __Free graphs as compact quantum metric spaces__

I will recall the construction of a "free graph algebra" and show how it forms a compact quantum metric space in the sense of Rieffel. Gromov-Hausdorff convergence properties of certain families of graphs will be discussed. This is joint work with Dave Penneys.

## 04:50-05:30 | **Michael Anshelevich:** __The exponential homomorphism in non-commutative probability__

The wrapping transformation is easily seen to intertwine convolutions of probability measures on the real line and the circle. It is also easily seen to not transform additive free convolution into the multiplicative one. However, we show that on a large class L of probability measures on the line, wrapping does transform not only the free but also Boolean and monotone convolutions into their multiplicative counterparts on the circle. This allows us to prove various identities between multiplicative convolutions by simple applications of the additive ones. The restriction of the wrapping to L has several other unexpected nice properties, for example preserving the number of atoms. This is joint work with Octavio Arizmendi. [Slides]

__Saturday, March 26__

## 09:00-09:30 | Coffee

## 09:30-10:10 | **Weihua Liu:** __On noncommutative distributional symmetries and de Finetti theorems associated with them__

I will present a recent work on general de Finetti theorems for classical, free and boolean independencies. I will give an application of our theorems to all easy quantum groups and easy groups. [Slides]

## 10:20-11:00 | **Tomohiro Hayase:** __De Finetti theorems for Boolean quantum semigroups__

We define a new kind of category $D$ of partitions and associated Boolean analogue of easy quantum groups $C(G_n^D)$. By going beyond the uniform case here, we prove a general form of Boolean de Finetti theorem. The theorem implies that the symmetry given by $(C(G_n^D))_{n \in \mathbb{N}}$ induces the conditional Boolean independence and an additional restriction on the distribution. [Slides]

## 11:10-11:30 | Coffee Break

## 11:30-12:10 | **Michael Brannan:** __Quantum channels arising from representations of free orthogonal quantum groups__

I will discuss some joint work with Benoit Collins, where we study quantum channels arising from finite dimensional representations of free orthogonal quantum groups. The main emphasis of this talk will be on how the underlying geometry of these quantum groups (non-amenability, property of rapid decay, planarity of the representation category etc...) allows one to construct interesting examples of (non-random) quantum channels.