SCIENTIFIC PROGRAMS AND ACTIVITIES

Todd Kemp, UC San Diego
The Large-$\mathbb{N}$ Limit of Brownian Motion on $\mathbb{GL_N}$

New Examples of Exchangeable Noncommutative Brownian Motions

We will begin by recalling a definition of noncommutative brownian motion due to Collins and Junge as well as a classification theorem of such brownian motions which satisfy an exchangeability condition. We will then discuss a couple of constructions which can be used to produce new examples of such exchangeable brownian motions. This is joint work with B. Collins and M. Junge.

Spectral and Brown measures of polynomials in free random variables

The combination of a selfadjoint linearization trick due to Greg Anderson with Voiculescu's subordination for operator-valued free convolutions and analytic mapping theory turns out to provide a method for finding the distribution of any selfadjoint polynomial in free variables. In this talk we will present the analytic machinery behind this process, and show an extension that allows in principle the computation of Brown measures of possibly non-selfadjoint polynomials in free variables. We will also indicate some results on Brown measures of some sums and products of free random variables. This talk is based on joint work with Tobias Mai and Roland Speicher and ongoing joint work with Piotr Sniady and Roland Speicher.

Strong asymptotic freeness for the generators of free orthogonal quantum groups

It is a well known fact that as $N$ tends to infinity, the standard coordinate functions over the $N \times N$ orthogonal group $O_N$ converge in distribution with respect to the Haar measure (after appropriate normalization) to a standard i.i.d. Gaussian family. In this talk, I will discuss the analogous situation that occurs in the large $N$ limit for Wang's free orthogonal quantum group $O_N^+$. In this case, T. Banica and B. Collins have shown that the limiting distribution of the normalized standard coordinate functions is Voiculescu's free semicircular system. We will show that in the free case, the mode of convergence to a free semicircular system is actually quite strong: Given any noncommutative polynomial in the normalized coordinates over $O_N^+$, not only does it converge in distribution to the corresponding polynomial over a free semicircular system, but also the norm of the polynomial converges to the norm of the limiting polynomial.

Cauchy-Stieltjes kernel families

The Cauchy-Stieltjes kernel families are constructed from a probability measure $\nu$ using the Cauchy kernel $x\mapsto 1/(1-\theta x)$ in a manner analogous to the celebrated exponential families that are constructed using kernel $x\mapsto\exp(\theta x)$. There are numerous similarities between the two families: both are parameterized by the mean, both are uniquely determined by the variance function (and the so called "domain of means"), and the variance function of the Cauchy-Stieltjes kernel family generated by the free convolution power of the generating measure $\nu$ has the same form as the variance function of the exponential family of the convolution powers.

There also some differences due to the fact that the exponential kernel $\exp(\theta x)$ is always positive while the Cauchy kernel $1/(1-\theta x)$ might be negative, and due to the fact that the variance of the Cauchy-Stieltjes kernel family might not exist.

The material on Cauchy-Stieltjes families comes from papers with Raouf Fakhfakh, Abgelhamid Hassairi, and Mourad Ismail.

A generalized Marchenko-Pastur Theorem

The well-known Marchenko-Pastur theorem gives the asymptotic spectral distribution of sums of random, independent, rank-one projections, when their size tends to infinity. Its main hypothesis is that these projections behave asymptotically as if they were uniformly distributed. We propose a way to drop this delocalization assumption and we generalize this theorem to a quite general framework, including random projections whose corresponding vectors are localized, i.e. with some components much larger than the other ones (joint work with Florent Benaych-Georges).

Transition operators for free convolution

By introducing a new universal space $\mathbb{C}\{X\}$ that extends the space $\mathbb{C}[X]$ of polynomials, we define transition operators which allows us to compute explicitly the conditional expectation of $P(A+B)$ given $B$, if $A$ and $B$ are free variables and $P\in \mathbb{C}[X]$. We investigate also the multivariate case and the multiplicative case. We finally describe some applications related to the free Segal-Bargmann transform introduced by Biane.

Comparison of L^p norms and free compression norms, and application to quantum information theory

We study and solve an optimization problem of the L^p norms with respect to the free compression norms. The solution to this problem clarifies and improves the violation of additivity for minimum output entropy in quantum information theory. This is joint work with Serban Belinschi and Ion Nechita.

Time reversal of free Stochastic Differential Equations and applications of non-microstates free entropy to von Neumann algebras

We show that solutions of free stochastic differential equations (SDE) with regular drifts and diffusion coefficients, when considered bacwards in time, still satisfy free SDEs for an explicit brownian motion. From a von Neumann algebra viewpoint, this implies that important subspaces of the L² space of the process, orthogonal to L²(W^*(X_t)), considered as a bimodules over the algebra of variables at time t W^*(X_t) are mixing and weakly contained in the coarse correspondence for almost every t. We give applications to free entropy, combining this information with previous joint results with Adrian Ioana using Popa's deformation/Rigidity Theory. We especially prove that finite non-microstates entropy implies non-Gamma and that finite non-microstates mutual information and extra assumptions (the main one being non-amenability of one algebra) implies primness.

Quantum symmetric states on universal free product C*-algebras

(joint work with Claus Koestler and John Williams)
We study quantum symmetric states on universal free product C*-algebras of a unital C*-algebra A with itself infinitely many times. This is a generalization of the notion of quantum exchangeable random variables, using the quantum permutation group of Wang. By extending and building on the proof of the noncommutative de Finetti theorem of Koestler and Speicher, we prove a de Finetti type theorem that characterizes quantum symmetric states in terms of amalgamated free products, with amalgamation over the tail algebra. This allows a convenient description of the set of all quantum symmetric states and in particular of the extreme quantum symmetric states. Finally, the sets of central quantum symmetric states and central tracial quantum symmetric states are shown to be Choquet simplexes.

On Generalisations of Free Probability Theory to Higher Dimensions

In this talk we will first discuss the algebraic structures underlying Free Probability Theory. Then based on this, we will introduce the generalisation of the S-transform to arbitrary dimensions. Whereas in the one-dimensional case it is abelian this is no longer true in dimensions bigger or equal to two. This has some important consequences, which we shall present, in particular also by examples from the analytic category.

Asymptotic Approximation and Inverse Problems for Sums of Free Variables

We shall discuss higher order expansions of the Voiculescu Entropy of sums of free variables under almost optimal moment conditions. Furthermore, results related to classical inverse characterization questions for linear forms of free variables are presented. This is joint work with G. Chistyakov.

An l^{p}-Version of von Neumann Dimension For Representations of Equivalence Relations

Following our previous ideas on extending von Neumann for Banach space representations of groups, we extend von Neumann dimension to representations of an equivalence relation on a Banach space. This extended dimension is defined in an "entropic" way, counting size of a certain set of microstates for the action of the equivalence realtion. Applications will include a way to define l^{p}-Betti numbers for equivalence relation, as well as giving some insight to the cost versus l^{2}-Betti number problem.

Free Real Algebraic Geometry and Free Convexity

Real Algebraic Geometry concerns polynomial inequalities and equalities. Say p >0 where q>0, then there is a algebraic relationship between p and q which is equivalent to this happening. Within the last decade there has been an effort to extend this sort of "algebraic certificate" to polynomials in (free) matrix variables. One important inequality a polynomial can satisfy is convexity. By now there is considerable theory of free convexity which continues to develop.

The talk will give an overview and describe the status of some parts of these two areas.

Supports of measures in a free additive convolution semigroup

Any Borel probability measure $\mu$ on $\mathbb{R}$ generates a discrete free additive convolution semigroup {$\mu^{\boxplus n}: n \in \mathbb{N}$}, which can be continuously imbedded into a partially defined semigroup {$\mu^{\boxplus n}: t \geq 1$}, as proposed by Bercovici and Voiculescu in their study of the free central limit theorem. In this talk, the supports and densities of measures in the continuous semigroup will be analyzed. More precisely, we will explain how the supports of the measures in the semigroup {$\mu^{\boxplus n}: t > 1$}vary when t increases. Moreover, motivated by the free central limit theorem we will give equivalent conditions so that the measure $\mu^{\boxplus n}$ has only one component in the support for sufficiently large t. An example of $\mu$ such that there are infinitely many components in the support of $\mu^{\boxplus n}$ for all t > 1 will be given as well.

The Large-$N$ Limit of Brownian Motion on $GL_N$

The Brownian motion on the Unitary group $U_N$ has a large-$N$ limit: the free unitary Brownian motion, introduced by Philippe Biane in 1997 and used extensively in free probability since then. In the same 1997 paper, Biane introduced a complex multiplicative free Brownian motion (as the solution of a free SDE) that he conjectured is the large-$N$ limit of the Brownian motion on $GL_N$. In this talk, I will discuss my recent work proving this conjecture, as well as classifying the large-$N$ limits of a class of diffusion processes on $GL_N$ that are invariant with respect to the real subgroup $U_N$ in a strong sense. The techniques rely, in part, on my recent joint work with Bruce Driver and Brian Hall on the large-$N$ limit of the unitary Segal--Bargmann transform; time permitting, I will discuss this as well.

Combinatorial relations between cumulants

We discuss combinatorial relations between various cumulants (classical, free, boolean and monotone) which we found in joint work with O.Arizmendi, T.Hasebe and C.Vargas.

Operator-valued free probability theory and the self-adjoint linearization trick

The linearization trick in its self-adjoint version by G. Anderson provides in many situations and particularly in free probability theory a powerful tool to deal with polynomials in non-commutative variables. It leads, for example, in combination with results about the operator-valued free additive convolution to an effective algorithm for the calculation of the distribution of any self-adjoint polynomial in free random variables (joint work with S. Belinschi and R. Speicher). In my talk, I will present some facts about the linearization trick and I will discuss some of its applications.

Some remarks on groups and cumulants

I describe how an interest in representations of certain finite simple groups led to free cumulants and ultimately the work with Roland Friedrich of which has spoke on Wednesday.
There are many open problems:
Are sporadic simples sporadic?
Does M have an action on the free loop space of the 24-dimensional manifold?
How general is moonshine?

Book: Mark Ronan, Symmetry and the Monster

Numerical Computation of Convolutions in Free Probability

We construct a numerical toolbox for computing the additive, multiplicative and compressive convolution operations from free probability. In the univalent case, the method exploits the fact that the output measure is, under broad conditions, either smooth and exponentially decaying (i.e. Schwartz) or an analytic function times a semicircle. High accuracy is obtain by representing the output measure using a basis that incorporate its regularity, and imposing that the inverse of the Cauchy transform takes the correct values at specified points in the complex plane. The approach is further extended to the multiple interval case, by taking into account the multivalued nature of the inverse of the Cauchy transform.

Rigidity for characters on lattices and commensurators

A character on a discrete group is a conjugation invariant function of positive type which take value 1 at the identity. The study of characters on infinite groups was initiated by Thoma in 1964 where he classified extreme characters on the group of finite permutations of the natural numbers. Recently, Bekka has shown that characters on the groups PSL_n(Z), for n > 2 have a remarkable rigidity property, they correspond either to finite dimension representations, or else the left regular representation. For the case n = 2, this was generalized to other rings of integers by the speaker and Thom. In this talk we will describe a similar rigidity property for characters on many irreducible lattices in higher rank semi-simple groups. This is related to rigidity results of Margulis, and Bader-Shalom, and confirms a conjecture of Connes for such groups. This is joint work with Darren Creutz.

Hardy classes on some non-commutative unit balls

The talk will present some results, joint work with V. Vinnikov, in the study of the Hardy space H2 of locally bounded non-commutative functions on the non-commutative unit ball of a variety of finite dimensional operator spaces. The results are part of a larger work on non-commutative symmetric domains. Our methods combine the general theory of non-commutative functions with asymptotic freeness results of D.-V. Voiculescu and formulas for integration over unitary groups of B. Collins and P. Sniady.

The Amalgamated Free Product of Semifinite Hyperfinite von Neumann Algebras over Type I atomic Subalgebras

The amalgamated free product for von Neumann algebras is a construction which has seen considerable use over the years. Here we examine the amalgamated free product of first finite then semifinite hyperfinite von Neumann algebras over type I atomic subalgebras. Generalising the concept of standard embeddings to what we call substandard embeddings we are able to show these are composed of direct sums interpolated free group factors and hyperfinite algebras in the finite case, and those with the addition of B(H) tensored with interpolated free group factors in the semifinite case (as well as allowing semifiniteness in the hyperfinite part). Further we show that these classes are closed under this type of amalgamated free product. We also adapt the concept of free dimension so that these products continue to satisfy the equation $fdim(A*_{D}B)=fdim(A)+fdim(B)-fdim(D)$.

Distributions of Polynomials of Free Semicircular Variables

One central concept in the study of free probability is to describe the spectral distributions of non-commutative polynomials of
freely independent random variables. In this talk we will briefly outline the recent results that if $\mu$ is the spectral distribution
of a self-adjoint (matricial) polynomial in freely independent semicircular variables then $\mu$ is non-atomic and the Cauchy
transform of $\mu$ is algebraic. The fact that $\mu$ is non-atomic follows from the extension of the strong Atiyah conjecture to certain tracial *-algebras and the fact that the Cauchy transform of $\mu$ is algebraic follows from demonstrating (using a specific Schwinger?Dyson equation) that a specific formal power series in non-commuting variables is algebraic. This is joint work with D. Shlyakhtenko.

Minimax theorems of Ky Fan and Wielandt

Let $\alpha_1 \leq \cdots \leq \alpha_n$ be an increasing rearrangement of the eigenvalues of a self-adjoint matrix in $M_n(\mathbb{C})$. The classical minimax theorem of Ky Fan is an extremal characterisation of $\sum_{i=1}^k \alpha_i$ for each $1 \leq k \leq n$. The classical minimax theorem of Wielandt is an extremal characterisation of $\sum_{j=1}^k \alpha_{i_j}$ for any subset $\{i_1, \cdots, i_k\} \subset \{1, \cdots, n\}$.
In joint work with Madhushree Basu, we have obtained `continuous versions' of Ky Fan's theorem as well as of Wielandt's theorem but only for $k=2$. Our attempts to prove this for all $k$ seem to be blocked by our proof depending on a result on `a standard form for two projections in a von Neumann algebra' which has no satisfactory analogue for more than two projections. An attempt will be made to describe our results and this hurdle as well as the reasons for our interest in Wielandt's theorem.

On the free Gamma distributions

In free probability theory the role of the Gaussian distribution is played by the semi-circle distribution, and the square of a semi-circular distributed random variable has the Marchenko-Pastur distribution. However, the Marchenko-Pastur distribution is also the free analog of the Poisson distribution, so the free analog of the chi-square distribution is instead thought of as the image of the classical chi-square distribution by the Bercovici-Pata bijection between the classes of infinitely divisible probability measures in classical and free probability. More generally the free gamma distributions are defined similarly in terms of the mentioned bijection. It follows from general properties of the Bercovici-Pata bijection that the free Gamma distributions are non-stable, selfdecomposable distributions (with respect to free convolution) with moments of any order, but further detailed properties remained until recently unrevealed. In the talk I will report on a recent detailed study of the free Gamma distributions based on the method of Stieltjes inversion. In particular I will address the following features: Absolute continuity, tail behavior and unimodality.
The talk is on joint work with Uffe Haagerup.

Orbital free entropy and its dimension counterpart

A couple of years ago, Hiai, Miyamoto and I introduced the orbital free entropy $\chi_\mathrm{orb}$ and its dimension counterpart $\delta_{0,\mathrm{orb}}$. The quantities originate in two things - the so-called change of variable formula by matrix diagonalization on the matrix Euclidean space on one hand and the free entropy adapted to projections studied by Hiai and myself following Voiculescu's proposal on the other hand. We will start with those origins, then report the current status of the study of those quantities, and finally conclude a few questions.

On the adjoint representation of orthogonal free quantum groups

I will review some operator-algebraic properties of free quantum groups and present a new result about the adjoint representation of orthogonal free quantum groups, with application to strong solidity.
This is joint work with Pierre Fima.

Realization and interpolation for noncommutative functions

I will discuss concepts, conjectures, and preliminary results generalizing the classical results on contractive analytic functions (or analytic functions with positive imaginary part) on the unit disc (or the upper halfplane) to the setting of noncommutative functions, that is functions defined on appropriate matrices of all sizes over a given vector space ${\mathcal V}$, which satisfy certain compatibility conditions as we vary the size of matrices --- they respect direct sums and similarities. The space ${\mathcal V}$ here is an operator space (or an operator system), so that we can define the noncommutative unit disc (or the noncommutative upper halfplane). In case ${\mathcal V}$ is finite dimensional, we recover essentially the results of Ball--Groenewald--Malakorn on the $d$-variable noncommutative Schur class whose elements are viewed as formal power series in $d$ noncommuting indeterminates.

This is a joint work in progress with Joe Ball, Serban Belinschi, and Mihai Popa.

Partitions and Quantum Groups

In 2009, Banica and Speicher initiated the systematic study of the so called easy quantum groups. These are compact matrix quantum groups with a very intrinsic combinatorial structure, given by partitions. The classification of these objects has recently been completed in joint work with Sven Raum, and I will report on that. Furthermore, the fusion rules (i. e. the representation theory) of easy quantum groups can be expressed in terms of partitions (joint work with Amaury Freslon). This gives a unified approach to the fusion rules of Wang's quantum permutation group S_n^+, the free orthogonal quantum group O_n^+, and many others.

Analytic Function Theory for Operator-Valued Free Probability

In this talk, we discuss the properties of analytic maps from the non-commutative upper-half plane to the non-commutative lower-half plane. In particular, we prove a characterization of the non-commutative Cauchy transforms that is analogous to the scalar-valued result. We will also discuss the immediate consequences of this work.

Applications of the time derivative of the $L^2$-Wasserstein distance and the free entropy dissipation.

For compactly supported probability measures on the real, we consider the time-evolution by the free Fokker-Planck equation. We shall see the time-derivative formula of the square of the $L^2$-Wasserstein distance by using the optimal transport map and the dissipation formulas of the free relative entropy. Then using these formulas we will show that the free analogue of HWI, the logarithmic Sobolev, and the transportation cost inequalities can be derived more directly for a strongly convex potential function.We shall also show the time derivative formula of the relative free entropy for the semicircular perturbations, which gives another integral representation of the relative free entropy in univariate case.

Free convolution with a free multiplicative analogue of the normal distribution

We obtain a formula for the density of the free convolution of an arbitrary probability measure on the unit circle
of $\mathbb{C}$ with the free multiplicative analogues of the normal distribution on the unit circle. This description relies on a characterization of the image of the unit disc under the subordination function, which also allows us to prove some regularity properties of the measures obtained in this way. We also obtain analogue results for probability measures on $\mathbb{R}^+$.