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, Anna Krystek (Politechnika Wrocławska), Romuald Lenczewski (Politechnika Wrocławska), Wojciech Młotkowski, Lahcen Oussi, Piotr Śniady, Anna Wysoczańska-Kula, Janusz Wysoczański

See also:

Current schedule:

Thursday, December 6, 2018, 10:15-12:00, room 604

Biswarup Das (Uniwersytet Wrocławski)

Admissibility conjecture for quantum group representations

A long-standing open problem in the representation theory of quantum groups is to decide whether the following statement is true or false: Every finite dimensional, unitary representation of (locally) compact quantum group ''factors'' in a suitable sense, through a representation of some matrix quantum group. This conjecture came up first in a work of P. Sołtan, and later on it kept featuring in many subsequent works on representation theory of (locally) compact quantum groups. We will give a partial solution to this conjecture, through proving that for a large class of (locally) compact quantum groups, this is true. Based on a joint work with P. Salmi and M. Daws.


Thursday, December 13, 2018, 10:15-12:00, room 604

Simon Schmidt (Universität des Saarlandes)

Quantum symmetries of finite graphs

The symmetry of a graph is captured by its automorphism group. This talk will concern a generalization of this concept in the framework of Woronowicz's compact matrix quantum groups, the so-called quantum automorphism group. A natural question is: When does a graph have no quantum symmetry, i.e. when does the quantum automorphism group coincide with the classical automorphism group? We will see that the Petersen graph has no quantum symmetry. Furthermore, we show that if the automorphism group of a graph contains a certain pair of automorphisms, this graph has quantum symmetry.



Previous talks:

Thursday, November 29, 2018, 10:15-12:00, room 604
Adrian Dacko (Politechnika Wrocławska)
V-monotone independence. Part III

Kontynuacja referatu z dnia 22 listopada.


Thursday, November 22, 2018, 10:15-12:00, room 604
Adrian Dacko (Politechnika Wrocławska)
V-monotone independence. Part II

Kontynuacja referatu z dnia 11 października.


Thursday, 8 November, 2018, 10:00-12:00, room 604
Adam Paszkiewicz (Uniwersytet Łódzki)
Fenomeny związane ze składaniem kontrakcji

Omówię paradoksy związane z trajektoriami $x_n := T_n \ldots T_1 x_0$ dla $n \in \mathbb{N}$, gdy $(T_1,T_2,\ldots)$ przebiega ciągi o wyrazach w skończonym zbiorze kontrakcji $\{Q_1,\ldots ,Q_k \}$, w ustalonej przestrzeni Hilberta \textit {H}. Silne metody dają twierdzenia o dylatacjach. Gdy $Q_1, \ldots ,Q_k$ muszą należeć do algebry von Neumanna $\mathcal{M}$, decydującą rolę odgrywa istnienie śladu skończonego $\tau$ na $\mathcal{M}$.


Thursday, 8 November, 2018, 10:00-12:00, room 604
Mitsuru Wilson (IM PAN)
Quantum symmetry of the toric noncommutative manifolds

In his framework, Rieffel showed that compact Lie groups of rank at least 2 admit nontrivial $\theta$-deformations as compact quantum groups. In my recent work, I showed that an action of such a Lie group $G$ on a manifold $M$ with a toric action can be extended to an action in the deformed setting. Of course, an action cannot be extended to an action of the deformed algebras for arbitrary $\theta$-parameters. First, I will explain what these deformations mean and I will explain exactly when an action in the classical setting extends to the noncommutative setting. I will also explain how the noncommutative 7-sphere S^7_Θ can be viewed as a quantum homogeneous space.


Thursday, October 25, 2018, 10:15-12:00, room 604
Patryk Pagacz (Uniwersytet Jagielloński)
Rozkłady typu Wolda dla stacjonarnych pól losowych oraz dla komutujących izometrii na przestrzeni Hilberta

Punktem wyjścia referatu jest twierdzenie Wolda, zarówno w pierwotnej wersji dotyczącej procesów stochastycznych jak i w abstrakcyjnej teorio-operatorowej wersji. Podczas referatu przyjrzymy się późniejszym analogicznym wynikom dotyczącym dwuwymiarowych pól losowych jak i niezależnie uzyskiwanym rezultatom dotyczącym rozkładów par izometrii na przestrzeni Hilberta. Przedstawimy znaczenie wyników, uzyskanych w ramach Teorii Operatorów, dla Procesów Stochastycznych i vice versa.


Thursday, October 11, 2018, 10:15-12:00, room 604
Adrian Dacko (Politechnika Wrocławska)
V-monotone independence

We introduce and study a new notion of noncommutative independence, called V-monotone independence, which generalizes the monotone independence of Muraki. We investigate the combinatorics of mixed moments of V-monotone random variables and prove the central limit theorem. We obtain a combinatorial formula for the limit moments and we find the solution of the differential equation for the moment generating function in the implicit form.


Thursday, October 4, 2018, 10:15-12:00, room 604
Alexander Bendikov (Uniwersytet Wrocławski)
On the spectrum of the hierarchical Schrödinger operator: the case of fast decreasing potential

This is the spectral analysis of the Schrödinger operator $H=L-V$ , the perturbation of the Taibleson-Vladimirov multiplier $L=D^{\alpha}$ by a potential $V$. Assuming that $V$ belonges to a class of fast decreasing potentials we show that the discrete part of the spectrum of $H$ may contain negative energies, it also appears in the spectral gaps of $L$. We will split the spectrum of $H$ in two parts: high energy part containing eigenvalues which correspond to the eigenfunctions located on the support of the potential $V$, and low energy part which lies in the spectrum of certain bounded Schrödinger operator acting on the Dyson hierarchical lattice. The spectral asymptotics strictly depend on the transience versus recurrence properties of the underlying hierarchical random walk. In the transient case we will prove results in spirit of CLR theory, for the recurrent case we will provide Bargmann's type asymptotics.



Seminar archives

Past schedule for the year 2017/2018
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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