# Past schedule for Discrete Harmonic Analysis Seminar 2006/2007

Seminar of the Mathematical Analysis Group in Institute of Mathematics, University of Wroclaw.
Current schedule available on www.math.uni.wroc.pl/analiza
Organizers: prof. Marek Bożejko (the main organizer), Piotr Śniady (technical organizer)
Time and place: usually every Wednesday, 10.15 - 12.00 in Institute of Mathematics, University of Wroclaw, room 607.

## Discrete harmonic analysis seminar

Topics: commutative and non-commutative harmonic analysis, quantum groups, combinatorics, quantum probability, free probability, Young diagrams, random matrices, convolutions of measures,....

Past schedule for the year 2004/2005
Past schedule for the year 2005/2006

People you may meet here: Marek Bożejko, Artur Buchholz, Ilona Królak, Anna Krystek, Romuald Lenczewski (Politechnika Wrocławska), Wojciech Młotkowski, Rafał Sałapata (Politechnika Wrocławska), Piotr Śniady, Łukasz Wojakowski, Janusz Wysoczański

List of all seminars in the Institute of Mathematics, University of Wrocław

# Current schedule available here

## Continuous interpolation between free and monotone gaussian operators

We introduce p-gaussian operators for continuous parameter p\in[0,1] which interpolates between free (p=1) and monotone (p=0) gaussian operatos. To descriptions of their moments we use a combinatorics of non-crossing ordered partitions related to the number of disorder. We also express 'Poisson moments' in terms of p-creation, p-anihilation and gauge operator.

# Previous talks:

Wednesday, 6 June, 2007, 10:15-12:00, room 603,

### Realization of Vershik--Kerov's type for finite factorial representations of infinite wreath product groups

We give a realization of arbitrary finite factorial representations of an infinite wreath product group, in which the role of classification parameters is visible. Our realization is an extension of a part of the Vershik--Kerov theory for the infinite symmetric group. This talk is based on a joint work with T.Hirai and E.Hirai.
Wednesday, 30 May, 2007, 10:15-12:00, room 603,

### Combinatorial relation between conditionally free cumulants and Jacobi coefficients---noncommutative case

Wednesday, 23 May, 2007, 10:15-12:00, room 603,

### Entanglement of positive-definite functions on compact groups

Starting from a separability problem of finite dimensional density matrices we reformulate it using positive definite functions on a product of compact groups. This allows us to generalize the notion of entanglement to group-theoretical setting. We prove an analog of Horodecki theorem which provides both necessary and sufficient criterion for entanglement of positive definite functions. It turns out that the role of positive maps is played by bounded operators on the space of continuous functions on the group which preserve positive definiteness.
Wednesday, 16 May, 2007, 10:15-12:00, room 603,

### bm-independence for algebras and general form of associated central limit theorem. Part II

We define a notion of bm-independence of algebras, and show how to generalize the classical Central Limit Theorem for bm-independent C*-algebras. This is a generalization of the previous result in which we treated special operators - so called bm-extension operators.
Wednesday, 9 May, 2007, 10:15-12:00, room 603,

### bm-independence for algebras and general form of associated central limit theorem. Part I

We define a notion of bm-independence of algebras, and show how to generalize the classical Central Limit Theorem for bm-independent C*-algebras. This is a generalization of the previous result in which we treated special operators - so called bm-extension operators.
Wednesday, 25 April, 2007, 10:15-12:00, room 603,

### Remarks on relations between Cauchy and Fourier transforms

The Cauchy transform of a probability measures is one of the main analytic tools is so called free-probability''. We will show how the Cauchy transforms, restricted to imaginary axis, can be view as a functional of (classical) Fourier transforms. In particular, this relation will allow to view boolean convolution as a transform of exponential mixtures. Main refernce: Cauchy transforms of measures viewed as some functionals of Fourier transforms'' , Probab. Math. Stat. vol. 26.1 (2006), pp. 187-200. (also available at http://www.math.uni.wroc.pl/~zjjurek/ )
Wednesday, 18 April, 2007, 10:15-12:00, room 603,

### Combinatorial relation between Jacobi coefficients and free cumulants

Wednesday, 11 April, 2007, 10:15-12:00, room 603,

### Miary półspektralne na prostej i ich splot boolowski i warunkowo wolny

Opowiem o podstawowych faktach dotyczących miar półspektralnych na prostej i podam ogolną definicję splotu Boolowskiego i warunkowo-wolnego takich miar. Przypadek miar o nośniku zwartym podał W.Młotkowski. Wyniki te otrzymalismy wspólnie z S.Belinschim. Zostaną też podane związki ze stanami koherentnymi i reprezentacją Bargmana grupy Heisenberga.
Wednesday, 4 April, 2007, 10:15-12:00, room 603,

### On a combinatorial explicit formula for representation of symmetric group

I will speak about a combinatorial formula, giving the caracter of irreducible representation of symmetric group on a short'' permutation. It has interesting consequences. First, it gives interesting results when the size of the diagram goes to infinity (upper bound for caracters, equivalent of some particular sequences). Second, it can be used to study Kerov polynomials. They are polynomials in some parameters of the shape of the diagram (the free cumulants) and their coefficient seems to have interesting combinatorial meanings.
Wednesday, 28 March, 2007, 9:00-10:15, room 603(?),

### Standard q-dilations

A q-dilation is introduced for a q-commuting row contraction is introduced and its relation with the minimal isometric dilation is studied. Some universal properties are obtained.
Wednesday, 28 March, 2007, 10:30-12:00, room 603,

### Kwantowe splatanie i odwzorowania dodatnie w algebrach macierzowych

Przedstawię elementarne wprowadzenie do zagadnień kwantowego splatania. Okazuje się, że zagadnienie to jest ściśle związane z teorią odwzorowań dodatnich. Jak wiadomo, w przeciwieństwie do odwzorowań całkowicie dodatnich, nie istnieje ogólny przepis konstrukcji odwzorowań dodatnich. W zastosowaniach fizycznych szczegolną rolę odgrywają odwzorowania dodatnie nierozkładalne. Znanych jest jedynie kilka klas takich odwzorowań. Zaprezentuję konstrukcje pewnej nowej klasy, która jest uogólnieniem pewnych znanych wcześniej przykładow.
Wednesday, 21 March, 2007, 10:15-12:00, room 603,

### Romuald Lenczewski (Wrocław University of Technology)

I will describe an operatorial approach to the theory of subordination in free probability. The analytic approach was developed by Voiculescu (1993) and Biane (1998) and until recently it was not known that also an operatorial approach can be given. In particular, I will show how to define suitable operators on the free product of Hilbert spaces which correspond to the subordination functions and their decompositions. This approach is based on the concepts of freeness with subordination' and orthogonal independence', which I introduced (2006) in the context of the subordination property for free additive convolutions.
Wednesday, 14 March, 2007, 10.15-12.00, room 603,

### Recent results in \mu-deformed Segal-Bargmann analysis (part 2)

We consider a \mu-deformation of the Segal-Bargmann transform, which is a unitary map from a \mu-deformed quantum configuration space onto a \mu-deformed quantum phase space (the \mu-deformed Segal-Bargmann space). Both of these Hilbert spaces have canonical orthonormal bases. We study L^p properties of this transform and obtain entropy-entropy inequalities (Hirschman) and entropy-energy inequalities (log-Sobolev) that generalize the corresponding known results in the undeformed theory. We obtain explicit formulas for the Shannon entropy of some of the elements of these bases.
Wednesday, 7 March, 2007, 10.15-12.00, room 603,

### Recent results in \mu-deformed Segal-Bargmann analysis (part 1)

We consider a \mu-deformation of the Segal-Bargmann transform, which is a unitary map from a \mu-deformed quantum configuration space onto a \mu-deformed quantum phase space (the \mu-deformed Segal-Bargmann space). Both of these Hilbert spaces have canonical orthonormal bases. We study L^p properties of this transform and obtain entropy-entropy inequalities (Hirschman) and entropy-energy inequalities (log-Sobolev) that generalize the corresponding known results in the undeformed theory. We obtain explicit formulas for the Shannon entropy of some of the elements of these bases.
Wednesday, 24 January, 2007, 10:15-12:00, room 607,

### Rafał Sałapata (Wrocław University of Technology)

Wednesday, 10 January, 2007, 10:15-12:00, room 607,

### New results concerning characters of symmetric groups S_n

I will present a conjecture of Stanley (math.CO/0606467) which gives a very nice formula for characters of symmetric groups S_n. This conjecture was proved by Feray (math.CO/0612090). I will show a number of equivalent formulations of this Stanley-Feray character formula. Depending on expectations of the audience I will present a new elementary proof of this result or present some applications (math.RT/0701051 - note that some proofs are not correct in the arXiv version).
Wednesday, 13 December, 2006, 10.15-12.00, room 607,

### Computations of free entropy dimension in hyperfinite case and in property T case (surveys on Jung's paper and on Jung and Shlyakhtenko's paper).

The seminar is a part of the course of my lectures titled Random matrices and free entropy''. The free entropy dimension \delta and its modified version \delta_0 were introduced by Voiculescu as some kinds of derivative of the free entropy \chi under the perturbation by semicircular motions. An important fact due to Kenley Jung is that \delta_0 is equal to the so-called fractal free entropy dimension \delta_1 defined in terms of covering/packing numbers of the set of approximating microstates. I will survey on the computation of \delta_0(X) in the cases where X=(X_1,\dots,X_n) generates a hyperfinite von Neumann algebra (due to Jung) and where X generates a property T von Neumann algebra (due to Jung and Shlyakhtenko).
Wednesday, 29 November, 2006, 10.15-12.00, room 607,

### On some precise deviation bounds in terms of weak parameters for vector valued Rademacher sums.

We establish a new deviation bound for Rademacher sums. The proof is short and completely elementary. As an application, we demonstrate that the optimal constants in Khinchine's and Khinchine-Kahane's inequalities are asymptotically the same.
Saturday, 25 November, 2006, 10.15-12.00,

### Application of Schur multipliers to means of Hilbert space operators

We associate Hilbert space operator means to given scalar means based on the characteriztion on Schur multipliers (together with the theory of Stieltjes double integral transformations. We obtain a handy criterion (in terms of the Fourier transformation) to check validity of norm comparison among operator means.
Wednesday, 15 November, 2006, 10.15-12.00, room WS,

### Harmonizable and Related Stochastic Processes on Abelian and Nonabelian Groups: An Operator-Space Approach

Let G be a locally compact group and X = {X_t, t\in G} be a stochastic process taking values in L^2_H = L^2(\Omega,A,P;H), the space of Bochner square-integrable functions on the probability space (\Omega,A,P) taking values in the Hilbert space H. If G is abelian with character group \hat{G} and X is stationary (in the wide sense), then the classical theorem of Bochner implies that the covariance function of X is given in terms of the Fourier-Stieltjes transform of a positive measure \mu on the diagonal of \hat{G}: < X_s,X_t>_H = \hat{\mu}(s - t) = \int_{\hat{G}} (s,\gamma), (t^{-1},\gamma) d\mu(\gamma). This leads to the existence of an L^2_H-valued stochastic measure'' whose Fourier transform is X and to many applications in the sciences. Similar results for nonabelian groups were obtained by Yaglom in the 1960's, which we shall describe. In this case the theory of unitary representations plays a role. In an effort to study nonstationary processes which preserve some analogues of the connection to the Fourier transform enjoyed by stationary processes on abelian groups, several notions of harmonizable'' processes were introduced over the years. For instance, one could drop the conditions that \mu be supported on the diagonal and positive. One goal of this program was to describe those processes for which a stochastic measure whose Fourier transform is X exists. Today a very satisfying anser to this question is available, and various characterizations and generalizations of this class of processes are known. Moreover, there are analogous results for processes parameterized by nonabelian groups G. The goal of this talk will be to describe these classes of processes and indicate how the burgeoning theory of operator spaces'' may be used to develop those characterizations easily for both abelian and nonabelian groups.
Wednesday, 18 October, 2006, 10.15-12.00, room 607,

### Complete decompositions of the free additive convolution and of the free product of graphs

I will introduce a new type of convolution of probability measures called the orthogonal convolution. Using this convolution, I will show how to derive alternating decompositions of the free additive convolution of compactly supported probability measures in free probability. These decompositions are directly related to alternating decompositions of the associated subordination functions and lead to an operator version of the analytic subordination. They also allow us to compute free additive convolutions of compactly supported measures without using R-transforms. In simple cases, representations of the associated Cauchy transforms as continued fractions are obtained in a natural way. Moreover, this approach establishes relations between convolutions and products associated with the main notions of noncommutative independence (free, monotone and boolean). Finally, this result leads to natural decompositions of the free product of rooted graphs.
Wednesday, 11 October, 2006, 10.15-12.00, room 607,

### Complete decompositions of the free additive convolution and of the free product of graphs

We define the following kind of operator on the vector space (C^n)^{\otimes N}: Let A_{\lambda} be a random Hermitian matrix, whose vector of eigenvalues is \lambda, and whose distribution is U\(n)-invariant. Define E_{\lambda}=\E(A_{\lambda}^{\otimes N}). The space (\C^{n})^{\otimes N} decomposes into an orthogonal sum of sectors, defined by the standard action of the groups U(n) and S_N. The operator E_{\lambda} is a linear combination of projections onto sectors. The Keyl-Werner theorem states, which kind of sectors take most of the mass in this combination. This allows us to use the operator E_{\lambda} for the study of random large partitions. Some applications and probabilistic interpretations will be discussed. In particular, we will explain the connection to a random process of \emph{growing partitions} studied by O'Connell. The necessary notions will be explained in the lecture.