## Prof. Dr. Roland Speicher

## Prof. Dr. Moritz Weber

## Oberseminar zur Freien Wahrscheinlichkeitstheorie

(Wintersemester 2016/2017)In diesem Seminar behandeln wir Themen aus der aktuellen Forschung zur Freien Wahrscheinlichkeitstheorie.

### Zeit und Ort

Mittwochs, 14-16 Uhr, SR 6### Vorträge

Zeit: 14:00 Uhr c.t. Ort: Seminarraum 6, Geb. E2 4

*Cumulants in Free Probability of type B*

Free Probability of Type B was introduced by Biane, Goodman, and Nica (2003) in order to create an analogous to Free Probability Theory, where the lattices of non-crossing partitions are replaced by the lattices of non-crossing partitions of type B. In this talk, we will see how this was done by the authors. More specifically, we will see how they use boxed convolution as a bridge between Cayley graphs of Symmetric groups and Free Probability Theory. Then, replacing Symmetric groups by Hyperoctahedral groups, they define cumulants of type B and a notion of free independence which is characterized by vanishing mixed cumulants of type B

Zeit: 15:00 Uhr s.t. Ort: Seminarraum 6, Geb. E2 4

*On the Rate of convergence for the Boolean and Monotone CLT*

The Classical Berry-Esseen Theorem states that the rate of convergence in the Central Limit Theorem, on the Kolmogorov metric, is of order 1/sqrt(n). An analogous result for the free CLT was obtained by Kargin in 2007 for measures of bounded support, being the rate of convergence also 1/sqrt(n). In 2008 Chystiakov and Götze improved that result only requiring that the fourth moment exists. In this talk I will speak about some results on the rate of convergence for the Boolean and monotone Central Limit Theorems.

Zeit: 14:00 Uhr s.t. Ort: Seminarraum 6, Geb. E2 4

*Spectral Analysis of Growing Graphs and Quantum Probability*

We talk about spectral analysis of large graph (or of a growing graph). We show how different products of graph are related to notions of independence. We also describe asymptotic distribution of distance-k graph of products. Finally we show how quantum probabilistic techniques are applied for the study of asymptotics of spectral distributions

Zeit: 14:00 Uhr c.t. Ort: Seminarraum 6, Geb. E2 4

*The classification of two-coloured partition quantum groups*

Building on ideas of T. Banica and R. Speicher, one can associate to a suitable collection of partitions with colours a compact quantum group. This is a rich source of examples which covers most of the quantum groups studied so far. It is therefore an interesting problem to classify the objects arising from this construction. Up to now, the strategy to prove classification results is to use combinatorial arguments to list the possible sets of partitions, independently of the associated quantum group. In this talk I will explain another strategy relying on the representation theory of the quantum groups and replacing most combinatorial arguments by algebraic computations in fusion rings. As an application, I will give a classification of all partition quantum groups on two slef-inverse colours. Together with the works of M. Weber, S. Raum and P. Tarrago this completes the classification of all partition quantum groups on two colours.

Zeit: 16:00 Uhr c.t. Ort: Seminarraum 10, Geb. E2 4

*Free component analysis*

We describe a method for unmixing mixtures of 'freely' independent random variables in a manner analogous to the indepedent component analysis (ICA) based method for unmixing independent random variables from their additive mixture. Random matrices play the role of free random variables in this context so the method we develop, which we call Free component analysis (FCA), unmixes matrices from an additive mixture of matrices. We describe the theory -- the various 'contrast functions', computational methods and compare FCA to ICA on data derived from real-world experiments.

Zeit: 15:00 Uhr s.t. Ort: Seminarraum 6, Geb. E2 4

*Non-singular Bernoulli actions and L^2-cohomology*

The study of probability preserving Bernoulli actions of discrete groups has a rich history in ergodic theory, dating back to von Neumann and Kolmogorov. However, as soon as one steps away from the probability preserving case, the results on such actions are scarce in spite of their prototypical nature. In fact, examples of Bernoulli actions of type III have only be given recently by Kosloff and Danilenko/Lemanczyk and only for the group of integers. I will present a joint article with Stefaan Vaes wherein we prove the following conjecture for almost all infinite discrete groups: An infinite discrete group admits a Bernoulli action of type III if and only if its first L^2-cohomology is non-zero.

Zeit: 16:00 Uhr c.t. Ort: Seminarraum 6, Geb. E2 4

*Some remarks on the facets of second-order free probability*

Extending a phrase by J. Mingo and R. Speicher [1], free probability has at least four basic facets: operator algebras, free harmonic analysis, random matrix theory, and combinatorics. In this talk we will discuss some work in progress with Jamie Mingo and Serban Belinschi concerning the random matrix theory and combinatoric facets of second-order free probability. As we will see, this approach may shed some light on problems concerning the other two facets.

[1] J. Mingo and R. Speicher. "Second order freeness and fluctuations of random matrices: I. Gaussian and Wishart matrices and cyclic Fock spaces." Journal of Functional Analysis, 2006.

Zeit: 14:00 Uhr c.t. Ort: Seminarraum 6, Geb. E2 4

*Non-hermitian random matrices with a variance profile*

For each n, let $A_n = (\sigma_{ij})$ be an $n\times n$ deterministic matrix and let $X_n = (X_{ij})$ be an $n\times n$ random matrix with i.i.d. centered entries of unit variance. We are interested in the asymptotic behavior of the empirical spectral distribution μYn of the rescaled entry-wise product $$ Y_n = \frac 1{\sqrt{n}}σ_{ij}X_{ij} . $$ For our main result, we provide a deterministic sequence of probability measures μn, each described by a family of Master Equations, such that the difference μYn −μn converges weakly in probability to the zero measure. A key feature of our results is to allow some of the entries σij to vanish, provided that the standard deviation profiles An satisfy a certain quantitative irreducibility property. This is a joint work with Nick Cook, Walid Hachem and David Renfrew.

Zeit: 14:00 Uhr c.t. Ort: Seminarraum 7, Geb. E2 4

*Multipartite rational functions: the universal skew field of fractions of a tensor product of free algebras*

A commutative ring embeds into a field if and only if it has no zero divisors; moreover, it this case it admits a unique field of fractions. On the other hand, the problem of localization of noncommutative rings and embeddings into skew fields (that is, division rings) is much more complex. For example, there exist noncommutative rings without zero divisors that do not admit embeddings into a skew field, and rings with several non-isomorphic "skew fields of fractions". This lead Paul Moritz Cohn to introduce the notion of the universal skew field of fractions to the general theory of skew fields in the 70's. However, almost all known examples of rings admitting universal skew fields of fractions belong to a relatively narrow family of Sylvester domains. One of the exceptions is the tensor product of free algebras. With the help of matrix evaluations we will construct the skew field of multipartite rational functions, which turns out to be the desired universal skew field of fractions. We will also explain its role in the difference-differential calculus in free analysis.

Zeit: 11:00 Uhr c.t. Ort: Seminarraum 6, Geb. E2 4

*Commuting operators in finite von Neumann algebras*

We find a joint spectral distribution measure for families of commuting elements of a finite von Neumann algebra. This generalizes the Brown measure for single operators. Furthermore, we find a lattice (based on Borel sets) consisting of hyperinvariant projections that decompose the spectral distribution measure. This leads to simultaneous upper triangularization results for commuting operators. (Joint work with Ian Charlesworth, Fedor Sukochev and Dmitriy Zanin.)

*Linearizing the Word Problem in (some) Free Fields*

We are interested in constructing minimal linear representations of elements in the universal field of fractions (free field) of the free associative algebra (over a commutative field). I will present a recent result which serves as a first step, namely solving the word problem with linear techniques and discuss some open problems. Since the factorization of non-commutative polynomials seems to be closely related, I try to sketch a possible setup.

Zeit: 10:30 Uhr s.t. Ort: Seminarraum 8, Geb. E2 4

*Products of Polynomial Ensembles*

In the past years a revival of products of random matrices has happened. This is due to the exact calculation of their joint probability densities of the eigenvalues as well as the singular values at finite matrix dimension and at a finite number of matrices multiplied. One question remained open why these exact calculations were possible. Kuijlaars et al. quite early identified a particular structure of the joint probability density functions which all these example of products as well as sums of random matrices satisfy. This class of ensembles is called polynomial ensembles. In a work of last year by Holger Koesters and me, we showed that the joint probability densities of the eigenvalues and the one of the singular values of a bi-unitarily invariant random matrix ensemble (for example the Ginibre ensemble) are bijectively related. As a byproduct of this result we could identify a proper subset of the polynomial ensembles. This set has the nice property that it builds a semi-group with respect to the multiplication of the corresponding bi-unitarily invariant random matrices and even has a semi-group action on the whole set of polynomial ensembles. I am going to talk about the consequences of this observation and a first but failed approach to analytically continue the number of matrices multiplied. I will also point out a way to circumvent the problems arising in such an analytical continuation.

*Rational function in strongly convergent random variables*

In this talk, we want to show that for a strongly convergent sequence of random variables, the convergence of trace and norm can be extended from polynomials to rational functions under certain assumptions. Roughly speaking, it claims that the strong convergence property is stable under taking the inverse of non-commutative random variables which have this property. As a direct corollary, we can conclude that for any rational function which is well-defined at a tuple of freely independent semi-circular elements, we have the convergence of trace and norm of this rational function in independent GUE random matrices to its evaluation at these semi-circular elements, almost surely. Such a result is expected in a recent paper by Helton, Mai and Speicher, where they can calculate the distribution of a rational expression in non-commutating random variables and their simulation indicates the convergence of corresponding random matrices should be true.

*Free cumulants and Schöder trees*

It is now well established that free cumulants, defined in terms of noncrossing partitions, give a combinatorial point of view on free probability. In the case of free cumulants of a single random variable, we can alternatively use the R-transform, an operation on functions that is essentially the compositional inversion. In this work we give an extension of the R-transform that covers the multi-variable case, using ideas coming the theory of operads, and an appropriate multilinear extension of free cumulants. We get in particular new combinatorial formulas involving Schröder trees rather than noncrossing partitions.

(Joint work with Frédéric Menous, Jean-Christophe Novelli, Jean-Yves Thibon).

*Stein kernels and the fourth moment theorem*

We will review some functional inequalities which are useful in order to measure the closedness of a random variable to the semicircular variable. It will allow us to give a quantitative version of the fourth moment theorem of Kemp-Nourdin-Peccati-Speicher.

This is joint work with Tobias Mai.

*An integration by parts on "space of loops"*

We consider to construct a measure on space of loops in $¥mathbb{C}^{*}$ (strictly speaking, a space of paths in a coefficient body), surrounding the origin by employing the utility of the (alternate) Loewner-Kufarev equation. We discuss about a simple integration by parts formula under the measure.

*Non commutative probability of type D*

We construct a deformed Fock space and a Brownian motion coming from Coxeter groups of type D. The construction is analogous to that of the $q$-Fock space (of type A) and the $(\alpha,q)$-Fock space (of type B).

*Potential applications of non-commutative probability to stochastic topology*

Recently, Drummond-Cole, Park and Terilla started a new branch of NCP by extending the notions of NC probability spaces to Homotopy probability spaces, in such a way that topological spaces become NC random variables and their distributions are evaluations of topological invariants.

In this talk, we give a brief introduction to the algebraic aspects of TDA (Topological Data Analysis) and we focus on the main topological invariants there, such as the Betti numbers. This suggests some specific directions in which the general and abstract framework of Homotopy Probability Spaces may be specialized.

*Asymptotics and sign of the quantum orthogonal Weingarten coefficients*

The quantum orthogonal Weingarten coefficients were introduced by Banica and Collins to compute the Haar measure on the free Orthogonal quantum group O_n^+, but little was known about them. Later, people including Banica, Collins, Curran, Speicher, gave asymptotics of these coefficients in some particular cases, and upper bounds in general. The main technical result presented in this talk is: (a) a formula for the leading term and decay of all coefficients (and in turn, a proof that they are generically non-zero), and (b) the fact that the Wg function is strictly monotone between its last pole and infinity, and in particular, never zero in this interval.

As a byproduct, this solves affirmatively an open question of Jones, asking whether all coefficients of the dual basis of the Temperley-Lieb algebra in the original algebra are non-zero. We will spend some time on the proof of the results, and on the initial motivation for the question.

This is joint work with Mike Brannan.

*The Waldspurger transform of permutations and alternating sign matrices*

In the mid 2000s Waldspurger and Meinrenken discovered new and interesting tilings of space, arising from the actions of finite and affine reflection groups. In order to study the combinatorics of these tilings for the symmetric group, we define the Waldspurger Transformation of a permutation. It turns out that the Waldspurger Transformation is well defined not just for permutations, but for any sum-symmetric matrix. We completely characterize the Waldspurger Transform of alternating sign matrices and show that this gives rise to a geometric realization of the hasse diagram for the alternating sign matrix poset-- the MacNeille completion of the Bruhat order.

*Planar algebra of a coaction and free wreath product*

In this talk, we will recall the work of Banica on the planar algebra canonically associated to the coaction of a compact quantum group. We will then relate this work to recent results on the description of intertwiner spaces of free wreath products, in order to abstractly define the free wreath product of two ergodic coactions. A Boolean decomposition of such a free wreath product will be given at the end of the talk.

This is a joint work with Jonas Wahl.

*On the Fluctuations of Block Gaussian Matrices*

About a decade ago, the fluctuations of the moments of Gaussian matrices were studied from a combinatorial point of view by Mingo, Nica and Speicher. In this talk, based on their results, we will study the fluctuations of the moments of block Gaussian matrices. In particular, we will find a close expression for a matricial version of the second-order Cauchy transform.

This is joint work with James Mingo and Serban Belinschi.

Aktualisiert am: 17. März 2017 Miguel Pluma | Impressum |