Discrete Harmonic Analysis and Noncommutative Probability

About the seminar

Seminar of the Mathematical Analysis Group in Institute of Mathematics, University of Wroclaw.

The main organizer: prof. Marek Bożejko

Time and place: every Thursday, 10.00 - 12.00 Institute of Mathematics, Wrocław University, room 604.

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

People you may meet here: Marek Bożejko (IMPAN), Biswarup Das, Wiktor Ejsmont (Uniwersytet Ekonomiczny), Rafał Kobuszewski, Anna Krystek (Politechnika Wrocławska), Romuald Lenczewski (Politechnika Wrocławska), Wojciech Młotkowski, Piotr Śniady, Anna Wysoczańska-Kula, Janusz Wysoczański

See also:

Current schedule:

Thursday, 22 February, 2018, 10:00-12:00, room 604

Piotr Śniady (IM PAN)

Jack polynomials and their (dual) combinatorics

Jack polynomials can be regarded as a deformation of Schur polynomials, a deformation which depends on one parameter. Their combinatorics was investigated by several prominent researchers, including RIchard Stanley. In 1996 Ian P. Goulden and David M. Jackson formulated several conjectures (which remain open until today) and which suggest that Jack polynomials are related to some deformation of the group algebras of the symmetric groups. During the talk I will present some recent results of Maciej Dołęga and Adam Burchardt.


Thursday, 1 March, 2018, 10:00-12:00, room 604

Alexandru Nica (University of Waterloo, Ontario, Canada)

Free probabilistic aspects of meandric systems

I will consider a family of diagrammatic objects (well-known to mathematical physicists and to combinatorialists) which go under the name of "meandric systems". I will explain how meandric systems arise in calculations done in a non-commutative probability framework, and I will show how tools from free probability can be used to study some aspects of the asymptotic behaviour of a random meandric system of large order. The talk is based on a joint work with Ian Goulden and Doron Puder (arXiv:1708.05188) and on a joint work with Ping Zhong (arXiv:1801.05501).


Previous talks:

Thursday, 8 February, 2018, 10:00-12:00, room 604

Wiktor Ejsmont

Wolne prawo tangensa

W referacie przedstawię centralne twierdzenie graniczne dla sumy wolnych komutatorów, które nazwaliśmy wolnym twierdzeniem tangensa.


Thursday, 25 January, 2018, 10:00-12:00, room 604

Biswarup Das

Locally compact quantum group

The theory of Locally compact quantum group, as opposed to the theory of compact quantum group, is relatively recent. The beginning of the subject was a paper of S.L. Woronowicz in 1998, where he proposed a possible set of axioms for what could be thought of as a locally compact quantum group. However, the only stumbling block of his theory was that it was not clear how to prove the existence of Haar weights.

In 2001, Kustermans and Vaes proposed another viewpoint for studying locally compact quantum group by assuming the existence of Haar weights, and showed that there approach in particular is a special case of the more general framework of Woronowicz. Although classically they give locally compact groups, it remained as an open problem, whether the two approaches are same i.e. they in essence define the same object.

In this talk we will give an introduction to the subject of locally compact quantum group through working with some classical groups. We will not go into analytical details, this would be a gentle introduction to the major aspects of the subject and will prepare the ground for following talk on Property (T).


Thursday, 4 January, 2018, 10:00-12:00, room 604

Anna Wysoczańska-Kula

Projekcje w wolnej probabilistyce

Celem odczytu jest przedstawienie dowodu twierdzenia o wolnych projektorach z nieopublikowanej pracy K. Frankiewicza.

Uwaga! W związku z wyjazdami naukowymi członków seminarium spotkania 11.01 i 18.01 są odwołane.


Thursday, 14 December, 2017, 10:00-12:00, room 604

Marek Bożejko

Wolna probabilistyka i zwiazki z logika wielowartosciowa Lukasiewicza oraz entropią Tsallisa

Celem odczytu sa nastepujace sprawy:
1. Wolne niezależne projektory, przykłady w iloczynach wolnych grup i ich opis.
2. Elementy logiki Łukasiewicza.
3. q-Entropia Tsallisa.
4. Problemy.


Thursday, 7 December, 2017, 10:00-12:00, room 604

Biswarup Das

Kazhdhan property (T) for quantum groups: summary and new results. II

This will be a continuation of the talk delivered on November 23, 2017.


Thursday, 30 November, 2017, 10:00-12:00, room 604

Włodek Bryc (University of Cincinnati)

Rodziny miar z jądrem Cauchy'ego i wielomianową funkcją wariancji.

Rodziny miar z jadrem Cauchy'ego mają wiele własności podobnych do rodzin miar z jądrem wykładniczym. (W statystyce rodziny miar z jądrem wykładniczym noszą nazwę naturalnych rodzin wykładniczych - natural exponential families.) W obu przypadkach rodzina miar ma naturalną parametryzację przez średnią i jest jednoznacznie wyznaczona przez tzw. funkcję wariancji, czyli przez wariancję wyrażoną jako funkcja średniej. W referacie skoncentruję się na przypadku, gdy funkcja wariancji jest wielomianem stopnia 3. Konstrukcja takiej rodziny korzysta z analitycznych metod wolnej probabilistyki, w szczególności z multiplikatywnej wolnej potęgi miary Marchenko-Pastura z prac Wojciecha Młotkowskiego i jego współautorów. Są to wyniki wspólne z Raouf Fakhfakh (Sfax, Tunezja) i Wojciechem Młotkowskim (Wrocław)


Thursday, 23 November, 2017, 10:00-12:00, room 604

Biswarup Das

Kazhdhan property (T) for quantum groups: summary and new results.

Property (T) was introduced in the mid 60’s by D. Kazhdan, as a tool to demonstrate that a large class of lattices are finitely generated. The discovery of Property (T) was a cornerstone in group theory and the last decade saw its importance in many different subjects like ergodic theory, abstract harmonic analysis, operator algebra and some of the very recent topics like C*-tensor categories. In the late 1980’s the subject of operator algebraic quantum groups gained prominence starting with the seminal work of Woronowicz, followed by works of Baaj, Skandalis, Woronowicz, Kustermans, Vaes and others. Quantum groups can be looked upon as noncommutative analogues of locally compact groups and in this sense it was quite natural to explore the possibility of extending the notion of Property (T) to the realm of quantum groups. This was done in the following sequence: Property (T) was first studied within the framework of Kac algebras (a precursor to the theory of locally compact quantum groups), then for algebraic quantum groups and discrete quantum groups, and finally for locally compact quantum groups by Joita, Petrescu, Fima, Soltan, Kyed, Skalski, Viselter, Daws, Brannan and Kerr. Thus far, Property (T) was studied only for quantum groups with trivial scaling automorphism group. Quite recently we found a way of extending all the above studies to quantum groups with non-trivial scaling automorphisms. This talk will be a summary of the results that we have. We will start from the situation with groups abd then move over gradually to the quantum groups. Based on an ongoing joint work with Pekka Salmi.


Thursday, 16 November, 2017, 10:00-12:00, room 604

Uwe Franz (Universite de Franche-Comte, Francja)

On monotone independence and quantum stochastic processes with monotonically independent increments

We will review the current knowledge about monotone convolutions, monotone infinite divisibility, and monotone increment processes. The talk will close with several conjectures and open problems. Based on joint work with Takahiro Hasebe, Ikkei Hotta, and Sebastian Schleissinger.


Thursday, 9 November, 2017, 10:00-12:00, room 604

Malte Gerhold (Ernst-Moritz-Arndt-Universität Greifswald)

Bi-monotone independence

A new notion of independence for pairs of noncommutative random variables will be discussed. As for bi-freeness, which comes from a simultanious "left" and "right" realization of freeness on the free product of Hilbert spaces, bi-monotone independence comes from the simultanious study of monotone and antimonotone independence on the tensor product of Hilbert spaces. We will present a central limit theorem which admits a combinatorial description using bi-monotone pair partitions.


Thursday, 26 October, 2017, 10:00-12:00, room 604

Wojciech Młotkowski

Ujemna określoność metryki na grafie

Wiadomo, że dla niektórych grafów metryka $d(x,y)$ jest ujemnie określona (drzewa, grafy Cayleya grup Coxetera). Dla grafu spójnego $G=(V,E)$ definiujemy stałą $QEC(G)$ (,,quadratic embedding constant'') jako supremum wszystkich sum \[ \sum_{x,y\in V}d(x,y)f(x)f(y), \] gdzie $d(x,y)$ oznacza odległość $x$ od $y$ w grafie $G$, a $f$ przebiega wszystkie funkcje $f:V\to\mathbb{R}$ o nośniku skończonym takie, że $\sum_{x\in V}f(x)=0$ oraz $\sum_{x\in V}f(x)^2=1$. Ujemna określoność metryki jest równoważna temu, że $QEC(G)\le0$. W pracy badamy $QEC(G)$ dla gwiazdka produktów grafów. Są to wspólne wyniki z N. Obatą (Sendai).


Thursday, 19 October, 2017, 10:00-12:00, room 604

Maria Elena Griseta (University of Bari Aldo Moro)

Distributions on non-symmetric position operators on Weakly Monotone Fock Space

In this talk we investigate the distributions for sums of random variables $$x_i={a_i}^{-}+{a_i}^{†}+\lambda {a_i}^0,$$ where ${a_i}^{-}$, ${a_i}^{†}$ and ${a_i}^0$ denote respectively annihilation, creation and conservation operators on the Weakly Monotone Fock Space and $\lambda\in \mathbb{R}$. We start considering sum of position operators (case $\lambda=0$): after obtaining a recursive formula for the moments \[ \mu_{m,n} := \omega_{\Omega}\bigg(\bigg(\sum_{k=1}^{m}(A_{k}+A_{k}^{\dagger})\bigg)^{2n}\bigg) \] where $\Omega=1\oplus 0\oplus 0 \oplus 0\oplus \ldots$ is the vacuum vector and $\omega_{\Omega}(\cdot)=\left\langle \Omega,\cdot\Omega\right\rangle$ is the vacuum expectation, we calculate the Cauchy Transform of the distribution measure and the density for $m=2$. Finally we compute the distribution of a single random variable $x_i$ with respect to the vacuum state for the general case $\lambda\neq 0$.


Thursday, 13 October, 2017, 10:00-12:00, room 604

Anna Wysoczańska-Kula

Związki teorii kohomologii z procesami Levy'ego na zwartych grupach kwantowych

W referacie opowiem o nieprzemiennych procesach Levy'ego definiowanych na grupach kwantowych i próbach ich klasyfikacji. Jedną z nich jest badanie istnienia rozkładu typu Levy'ego-Chinczyna. Okazuje się, że istnienie takiego rozkładu jest zagwarantowane przez pewne warunki kohomologiczne (choć nie jest im równoważne). Do zobrazowania opisanej teorii posłużą uniwersalne grupy kwantowe.


Thursday, 28 September, 2017 10:15-12:00, room 604

Wiktor Ejsmont

Cancellation phenomenon in free probability II

I am going to continue my previous discussion about the cancellation phenomenon of sample variance.



Past schedule for the year 2016/2017
Past schedule for the year 2015/2016
Past schedule for the year 2014/2015
Past schedule for the year 2013/2014
Past schedule for the year 2012/2013
Past schedule for the year 2011/2012
Past schedule for the year 2010/2011
Past schedule for the year 2009/2010
Past schedule for the year 2008/2009
Past schedule for the year 2007/2008
Past schedule for the year 2006/2007
Past schedule for the year 2005/2006
Past schedule for the year 2004/2005

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