Thursday, June 13, 2019, 10:15-12:00, room 604 David Shoikhet (The Galilee Research Center for Applied Mathematics & Holon Institute of Technology) Resolvent method in complex analysis and geometric function theory
Historically, complex dynamics and geometrical function theory have been intensively developed from the beginning of the twentieth century. They provide the foundations for broad areas of mathematics. In the last fifty years the theory of holomorphic mappings on complex spaces has been studied by many mathematicians with many applications to nonlinear analysis, functional analysis, differential equations, classical and quantum mechanics. The laws of dynamics are usually presented as equations of motion which are written in the abstract form of a dynamical system: ((dx)/(dt))+f(x)=0, where x is a variable describing the state of the system under study, and f is a vector-function of x. The study of such systems when f is a monotone or an accretive (generally nonlinear) operator on the underlying space has been recently the subject of much research by analysts working on quite a variety of interesting topics, including boundary value problems, integral equations and evolution problems. There is a long history associated with the problem on iterating holomorphic mappings and their fixed points, the work of G. Julia, J. Wolff and C. Carathéodory being among the most important. In this talk we give a brief description of the classical statements which combine celebrated Julia's Theorem in 1920 , Carathéodory's contribution in 1929 and Wolff's boundary version of the Schwarz Lemma in 1926 and their modern interpretations. Also we present some applications of complex dynamical systems to geometry of domains in complex spaces and operator theory.
Thursday, June 6, 2019, 10:15-12:00, room 604 Jakub Gismatullin (Uniwersytet Wroclawski) Metric ultraproducts of groups: simplicity and amenability
My talk will be about groups equipped with bi-invariant metrics/norms. I will explain, in elementary terms, both uniform metric amenability and uniform metric simplicity of groups; including examples and potential applications. These generalize, respectively, the previously studied notions of uniform amenability and uniform simplicity.
Thursday, May 30, 2019, 10:15-12:00, room 604 Isabelle Baraquin (Université de Franche-Comté) Analysis and probability on finite quantum groups and dual groups
In this talk, we will first present a result from Diaconis, Shahshahani and Evans. Let M be a random matrix chosen from the unitary group $U(n)$ and distributed according to the Haar measure. Then, for $j\in \mathbb N$, ${\rm Tr}(M^j)$ are independent and distributed as some complex Gaussian random variables when $n\to \infty$. We will look at this type of result in the framework of finite quantum groups and then in the unitary dual group.
The talk is a part of the Polonium programme, co-financed by the Polish National Agency for Academic Exchange (NAWA).
Thursday, May 23, 2019, 10:15-12:00, room 604 Wojbor Woyczyński (Case Western Reserve University) Multiscale conservation laws driven by Levy stable and Linnik diffusions: asymptotics, explicit representations, shock creation, preservation and dissolution
We will discuss the interplay between the nonlinear and nonlocal components of the evolution equations. In the particular case of supercritical multifractal conservation laws (CL) the asymptotic behavior, as $t \neq 1$, is dictated by the linearized case. For $\alpha < 1 $, the equations driven by infinitesimal generators of Levy -stable diffusions the solution exhibit shocks (for bounded, odd, and convex on $\mathbb R^+$, initial data) which disappear over a nite time. For Levy -Linnik diffusions, $0 < \alpha < 2$ , the local behavior is strikingly different. The relevant CLs display shocks that do not dissipate over time.
Monday, May 20, 2019, 10:15-12:00, room 603 Eugene Lytvynov (Swansea University) Fock representations of multicomponent commutation relations
Let $H$ be a separable Hilbert space and $T$ be a self-adjoint bounded linear operator on $H^{\otimes 2}$ with norm $\le1$, satisfying the Yang-Baxter equation. Bożejko and Speicher (1994) proved that the operator $T$ determines a $T$-deformed Fock space $\mathcal F(H)=\bigoplus_{n=0}^\infty\mathcal F_n(H)$. We start with reviewing and extending the known results about the structure of the $n$-particle spaces $\mathcal F_n(H)$ and the commutation relations satisfied by the corresponding creation and annihilation operators acting on $\mathcal F(H)$. We then choose $H=L^2(X\to V)$, the $L^2$-space of $V$-valued functions on $X$. Here $X:=\mathbb R^d$ and $V:=\mathbb C^m$ with $m\ge2$. Furthermore, we assume that the operator $T$ acting on $H^{\otimes 2}=L^2(X^2\to V^{\otimes 2})$ is given by $(Tf^{(2)})(x,y)=C_{x,y}f^{(2)}(y,x)$. Here, for a.a. $(x,y)\in X^2$, $C_{x,y}$ is a linear operator on $V^{\otimes 2}$ with norm $\le1$ that satisfies $C_{x,y}^*=C_{y,x}$ and the spectral quantum Yang-Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function $C_{xy}$ in the case $d=2$ is associated with plektons. For a multicomponent system, we describe its $T$-deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems. This is joint work with A. Daletskii, A. Kalyuzhny and D. Proskurin.
Thursday, May 16, 2019, 10:15-12:00, room 604 Eugene Lytvynov (Swansea University) Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis
For certain Sheffer sequences $(s_n)_{n=0}^\infty$ on $\mathbb C$, Grabiner (1988) proved that, for each $\alpha\in[0,1]$, the corresponding Sheffer operator $z^n\mapsto s_n(z)$ extends to a linear self-homeomorphism of $\mathcal E^{\alpha}_{\mathrm{min}}(\mathbb C)$, the Fr\'echet topological space of entire functions of exponential order $\alpha$ and minimal type. In particular, every function $f\in \mathcal E^{\alpha}_{\mathrm{min}}(\mathbb C)$ admits a unique decomposition $f(z)=\sum_{n=0}^\infty c_n s_n(z)$, and the series converges in the topology of $\mathcal E^{\alpha}_{\mathrm{min}}(\mathbb C)$. Within the context of a complex nuclear space $\Phi$ and its dual space $\Phi'$, in this work we generalize Grabiner's result to the case of Sheffer operators corresponding to Sheffer sequences on $\Phi'$. In particular, for $\Phi=\Phi'=\mathbb C^n$ with $n\ge2$, we obtain the multivariate extension of Grabiner's theorem. Furthermore, for an Appell sequence on a general co-nuclear space $\Phi'$, we find a sufficient condition for the corresponding Sheffer operator to extend to a linear self-homeomorphism of $\mathcal E^{\alpha}_{\mathrm{min}}(\Phi')$ when $\alpha>1$. The latter result is new even in the one-dimensional case.
Friday 10th – Saturday 11th May 2019, room EM
Hartman Special Seminar on Harmonic Analysis
Two-days meeting commemorating the 50 years history of Professor Hartman’s Seminar
Thursday, April 18, 2019, 10:15-12:00, room 604 Paweł Józiak (Politechnika Warszawska) On quantum increasing sequences (again)
Quantum increasing sequences were introduced by S. Curran to characterize free amalgamated independence of an infinite sequence of random variables (over the tail algebra) by means of the so-called quantum spreadability. This is a de Finetti type theorem that in the classical case was first established by C. Ryll-Nardzewski, but the assumptions are formally weaker. The plan of the talk is to discuss the connection of this structure to quantum permutation groups and describe in full generality the quantum subgroups generated by them. This is done by establishing a certain inductive-type framework for generation in quantum groups, a combinatorial description of quantum permutation groups by means of Temperley-Lieb diagrams and recent resolution of Banica's conjecture: there is no intermediate quantum group satisfying $S_5\subset\mathbb{G}\subset S_5^+$.
Thursday, April 11, 2019, 10:15-12:00, room 604 Biswarup Das (Uniwersytet Wrocławski) TBA
Thursday, April 4, 2019, 10:15-12:00, room 604 Wojciech Młotkowski (Uniwersytet Wrocławski) Rodzina ciągów typu Catalana
Omówię pewną rodzinę ciągów które są zdefiniowane w podobny sposób jak liczby Catalana. Dla takich ciągów badamy odpowiednie funkcje tworzące, dodatnią określoność oraz związki z wolną probabilistyką. Referat oparty jest na wspólnej pracy z Elżbietą Liszewską.
Thursday, March 28, 2019, 10:15-12:00, room 604 Romuald Lenczewski (Politechnika Wrocławska) Warunkowo monotoniczna niezależność i związany z nią produkt grafów
Kontynuacja referatu z 21.03.
Thursday, March 21, 2019, 10:15-12:00, room 604 Romuald Lenczewski (Politechnika Wrocławska) Warunkowo monotoniczna niezależność i związany z nią produkt grafów
Przedstawię konstrukcję, która sprowadza niezależność warunkowo monotoniczną Hasebe do niezależności tensorowej. Wynik ten zastosuję do zdefiniowania nowego produktu grafów, związanego z warunkowo monotonicznym splotem miar probabilistycznych ma prostej.
Thursday, March 7, 2019, 10:15-12:00, room 604 Biswarup Das (Uniwersytet Wrocławski) Hudson-Parthasarathy (HP) dilation of translation invariant Markov semigroups on CQG algebras
Continuation of the talk delivered on February 28.
Thursday, February 28, 2019, 10:15-12:00, room 604 Biswarup Das (Uniwersytet Wrocławski) Hudson-Parthasarathy (HP) dilation of translation invariant Markov semigroups on CQG algebras
In the first part of this talk, we will briefly review the theory of Hudson-Parthasarathy (HP) stochastic calculus and stochastic dilation of Markov semigroups in general. In the second half, we will consider CQG algebras and we will show that any translation invariant Markov semigroup on a CQG algebra can be stochastically dilated using Hudson-Parthasarahty calculus. Based on a joint work with Martin Lindsay.
Thursday, February 21, 2019, 10:15-12:00, room 604 Jacek Małecki (Politechnika Wrocławska) Zbieżność miar empirycznych dla rozwiązań stochastycznych macierzowych równań różniczkowych
Będziemy rozważać procesy miar empirycznych związanych z wartościami własnymi rozwiązań bardzo ogólnych macierzowych równań różniczkowych. Zbadamy ich zachowanie asymptotyczne w przypadku, gdy rozmiar macierzy rośnie do nieskończoności. Pokażemy ciasność rozważanych rodzin miar przy bardzo ogólnych założeniach na współczynniki odpowiadających im stochastycznych równań różniczkowych. Scharakteryzujemy rozkłady graniczne słabo zbieżnych podciągów jako rozwiązania równań całkowych. Wyniki te posłużą do wyznaczenia klas uniwersalności dla macierzowych procesów stochastycznych. Pokażemy także pewne nowe zjawiska tj. istnienie uogólnionych rozkładów Marchenko-Pastura o nośniku na całej prostej. Wskażemy związek otrzymanych wyników z wolną probabilistyką dowodząc zbieżności badanych obiektów do wolnych dyfuzji. Wyniki pochodzą ze wspólnej pracy z José Luisem Pérezem (CIMAT, Meksyk). Preprint pracy można znaleźć na arXiv pod poniższym linkiem https://arxiv.org/abs/1901.02841.
Thursday, February 14, 2019, 10:15-12:00, room 604 Wiktor Ejsmont (Uniwersytet Wrocławski) Generalized Gaussian processes and realizations on some Fock spaces
In my talk we will consider generalized processes associated with q-Meixner -Pollaczek orthogonal polynomials.
Thursday, February 7, 2019, 10:15-12:00, room 604 Louis Labuschagne (North-West University, RPA) Entropy for general quantum systems
We revisit the notion of relative entropy for both classical and quantum systems, and provide some new descriptions of this notion, respectively based on the theories of the Connes cocycle derivative, and noncommutative $L^p$-spaces. We then introduce the notion of entropy for a single state of a general quantum system, and show that this notion agrees with von Neumann entropy in the case of semifinite von Neumann algebras. In closing we investigate the relationship between this notion of entropy and relative entropy, and identify an Orlicz space which forms the home for all states with ``good'' entropy. This is a joint work with Adam Majewski.
Thursday, January 31, 2019, 10:15-12:00, room 604 Ewa Damek (Uniwersytet Wrocławski) Solutions to smoothing equations - existence of density.
We study the stochastic equation $$Y=_{law}\sum _{j=1}^N T_jY_j, \quad \quad \qquad (1) $$ where $=_{law}$ denotes equality in law, $(T_1,T_2,....)$ is a given sequence of complex variables i.e. random variables with values in $\mathbb{C}$ and $Y_1,Y_2,...$ are independent copies of the complex random variable $Y$ and independent of $(T_1,T_2,....)$. Let $N$ be the (possibly random) number of $T_j$ that are non zero. As long as $\mathbb{E}N>1$, the law of $\sum _{j= 1}^N T_jY_j$ behaves much better with respect to local regularity than the law of $Y$ which justifies the name ``smoothing''. For the real valued case and $T_j\geq 0$ the first result about absolute continuity of $Y$ was obtained by Liu. Recently, under suitable assumptions on $(T_1,T_2,....)$, a complete description of the set of solutions to (1) has been provided by Meiners and Mentemeier (2017). Using that we (Damek and Mentemeier) have proved absolute continuity of $Y $ in the complex case. Limit theorems for a broad class of random recursive structures and algorithms are derived by a ``so called'' contraction method. It was introduced by R\"osler in 1991 for the distributional analysis of complexity of Quicksort and later on has been extended to a variety of random recursive structures like recursive algorithms, data structures, Polya urn models and random tree models. Limit distributions derived by the contraction method are given implicitly as solutions to (1). Often there is no explicit formula for the distribution of $Y$ but still many properties of these limits may be derived from the equation. One of them is the absolute continuity, possibly differentiability which is important for the sake of possible applications.
Thursday, January 23, 2019, 10:15-12:00, room 604 Xumin Wang (Universite Bourgogne Franche Comte) Invariant Markov semigroups on the quantum homogeneous spaces
We obtained some one-to-one correspondences of invariant quantum Markov semigroups acting on expected coideal *-subalgebras and certain convolution semigroups of states on the underlying compact quantum group. This gives an approach to classifying invariant quantum Markov semigroups on these quantum homogeneous spaces. The classical sphere $S^{N-1}$, the free sphere $S^{N-1}_+$, and the half-liberated sphere $S^{N-1}_*$ are considered as examples. On these spheres, we classifying the generators of the invariant quantum Markov semigroups by calculating their eigenvalues and eigenvector spaces. The generators obtained in this way can naturally be viewed as Laplace operators on these spaces. The talk is a part of the Polonium programme, co-financed by the Polish National Agency for Academic Exchange (NAWA).
Thursday, January 3, 2019, 10:15-12:00, room 604 Jakub Gismatullin (IM PAN & UWr) On counting multifurcating trees and metric ultraproducts of groups
In the first part I will present work in progress with A. Żywot on counting rooted phylogenetic multifurcating trees. We study recurrence $T_{n,m}=(n+m-2)T_{n-1,m-1} + m T_{n-1,m}$, $n>m>1$, due to J. Felsenstein, where $T_{n,m}$ is the number of all such trees trees with $n$ leaves and $m$ internal leaves. After certain transformation this recurrence gives apparently new family of polynomials of binomial type. Generating functions and cumulants of it is closely related to Maclaurin expansion of $\frac{x^2}{\exp(x)-x-1}$, and maybe also to some generalized Bernoulli numbers. In the second part I will speak about groups with conjugacy-invariant norms and metric ultraproducts, focusing on stronger (uniform) versions of simplicity and amenability of metric ultraproducts of groups, for example uniform metric amenability, which is strictly weaker that uniform amenability and strictly stronger than amenability. I will also present a Foelner condition for amenability of topological groups (due to Thom and Schneider) in this context. This part is a joint work with M. Ziegler.
Thursday, January 10, 2019, 10:15-12:00, room 604 Ignacio Vergara (IM PAN) Radial Schur multipliers
A Schur multiplier on a set $X$ is a function $\varphi:X\times X\to\mathbb{C}$ defining a map $M_\varphi$ on the algebra of bounded operators on $\ell_2(X)$ by $\langle M_\varphi(T)\delta_y,\delta_x\rangle=\varphi(x,y)\langle T\delta_y,\delta_x\rangle$, for all $T\in\mathcal{B}(\ell_2(X))$. In this talk, I will focus on the particular case when $X$ is (the set of vertices of) a graph and the function $\varphi$ depends only on the distance between each pair of vertices. Such functions are said to be radial. For homogeneous trees, radial Schur multipliers were characterised by Haagerup, Steenstrup and Szwarc by the fact that a certain Hankel matrix belongs to the trace class. I will present some extensions of this result to larger classes of graphs: products of trees, CAT(0) cube complexes and products of hyperbolic graphs. The study of such graphs is motivated by some results on weak amenability for groups acting on them.
Thursday, January 17, 2019, 10:15-12:00, room 604 Marek Bożejko Central Limit Theorem, Generalized Gaussian processes and realizations on some Fock spaces
In my talk we will consider the following subjects: 1. Abstract CLT and connections with generalized Hermite polynomials like: (a) q-continuus (b) q-discrete (c) q-Meixner -Pollaczek (d) Free Meixner (Kesten) (e) arbitrary central orthogonal polynomial can be considered as generalized Hermite polynomial. 2. Connections with positive definite functions on permutation (hyperoctahedral) groups and Thoma characters on that class of groups.
Thursday, December 17, 2018, 14:00-15:30, room 604 Natasha Blitvic (Lancaster University) Noncommutative Central Limit Theorems
We will discuss noncommutative versions of the Central Limit Theorem. We will describe the general ideas behind such constructions, along with several recent examples and some current work in progress. Parts of this talk will be based on recent joint work with W. Ejsmont.
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.
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, 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.