About the seminar

Seminar of the Research Group in Mathematical Analysis (Zakład Analizy Matematycznej) in Institute of Mathematics, University of Wroclaw.

The main organizer: prof. Marek Bożejko (IMPAN)

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), Adrian Dacko, Biswarup Das, Wiktor Ejsmont, Anna Krystek (Politechnika Wrocławska), Romuald Lenczewski (Politechnika Wrocławska), Wojciech Młotkowski, Lahcen Oussi, Anna Wysoczańska-Kula, Janusz Wysoczański

Current seminar schedule:

Thursday, May 13, 2021 - 10:30, room zoom.us

Maciej Nowak

Eikonal formulation of large dynamical random matrix models"

The standard approach to dynamical random matrix models relies on the description of trajectories of eigenvalues. Using the analogy from optics, based on the duality between the Fermat principle (rays) and the Huygens principle (wavefronts), we formulate the Hamilton-Jacobi dynamics for large random matrix models, and we relate this dynamics to Voiculescu equation. The resulting formalism describes a broad class of random matrix models in a unified way, including normal (Hermitian or unitary) as well as strictly non-normal dynamics. HJ formalism applied to Brownian bridge dynamics allows one for calculations of the asymptotics of the Harish-Chandra-Itzykson-Zuber integrals.

Previous talks:

Thursday, May 13, 2021 - 10:30, room zoom.us

Maciej Nowak

Eikonal formulation of large dynamical random matrix models"

Thursday, May 6, 2021 - 11:00-12:00, room zoom.us

Octavio Arizmendi Echegaray

Energy of graphs and vertices"

Energy of graphs, introduced in mathematics by Gutman in the 70’s is a quite studied invariant of graphs. In (2018) with Juarez, we introduced a refinement called Energy of a Vertex. This allows to give better bounds and a local understanding for the energy of a graph. In this talk I will survey on the topic of Energy of Graphs and on recent results in relation with Energy of a Vertex, that I derived with various coauthors.

Thursday, May 6, 2021 - 10:30-11:00, room zoom.us

Mariusz Tobolski

Equivariant dimensions on graph C*-algebras (P2)"

Graph C*-algebras are universal C*-algebras associated with directed graphs that generalize Cuntz--Krieger algebras. Many examples of C*-algebras turn out to be graph C*-algebras, e.g. matrix algebras, the Toeplitz algebra, the Cuntz algebras, the C*-algebra of compact operators on a separable Hilbert space, q-deformed spheres, q-deformed projective spaces, q-deformed lens spaces. The advantage of this class of C*-algebras is that many of their properties, like simplicity or classification of certain ideals, can be described purely in terms of the underlying graph. There is a natural U(1)-action, called the gauge action, on every graph C*-algebra. In my talk, I will concentrate on this action and its restriction to finite subgroups from the perspective of the recently introduced local-triviality dimension of the action. I will illustrate all results in particular examples.

Thursday, April 29, 2021 - 10:30, room zoom.us

Mariusz Tobolski

Equivariant dimensions on graph C*-algebras"

Thursday, April 22, 2021 - 10:30, room zoom.us

Anatol Odzijewicz

Some Aspects of Positive Kernel Method of Quantization"

We discuss various aspects of positive kernel method of quantization of the one-parameter groups $\tau_{t} \in Aut(P, v)$ of automorphisms of a G-principal bundle $P(G, \pi, M)$ with a fixed connection form $v$ on its total space $P$. We show that the generator $\hat{F}$ of the unitary flow $U_{t}= e^{it\hat{F}}$ being the quantization of $\tau_{t}$ is realized by a generalized Kirillov-Kostant-Souriau operator whose domain consists of sections of some vector bundle over $M$, which are defined by suitable positive kernel on $P \times P$. This method of quantization applied to the case when $G = GL(N, \mathbb{C})$ and $M$ is a non-compact Riemann surface leads to quantization of the arbitrary holomorphic flow $\tau_{t}^{hol}\in Aut(P, v)$. For the above case, we present the integral decompositions of the positive kernels on $P × P$ invariant with respect to the flows $\tau_{t}^{hol}$ in terms of spectral measure of $\hat{F}$. These decompositions generalize the ones given by Bochner theorem for a positive kernels on $\mathbb{C}\times\mathbb{C}$ invariant with respect to the one-parameter groups of translations of complex plane.

Thursday, April 25, 2021 - 10:30, room zoom.us

Uwe Franz (Université de Franche-Comté)

De Finetti theorems for the unitary dual group (a.k.a. the Brown-Glockner-von Waldenfels algebra)

I will present several De Finetti theorems for the unitary dual group, also known as the Brown or Brown-Glockner-von Waldenfels algebra. This algebra, which is the universal *-algebra generated by the coefficients of a unitary, has the structure of a (*-algebraic) dual group (in the sense of Voiculescu) and also the structure of an involutive bialgebra. Therefore we can consider two different kinds of actions of this algebra on sequences of noncommutative random variables. Furthermore, the five universal notions of independence (as classified by Muraki) lead to fives different notions of invariance under the action as a dual group, which have different distributional characterizations. In particular, we show that a finite sequence of noncommutative random variables is invariant under the unitary dual group w.r.t. the free product iff it is composed of certain R-diagonal elements. Based on joint work with Isabelle Baraquin, Guillaume C\'ebron, Laura Maassen and Moritz Weber.

Thursday, April 08, 2021 - 10:30, room zoom.us

Adrian Dacko (Politechnika Wrocławska)

V-monotoniczne centralne twierdzenie graniczne, cz. 4

W referacie zostanie zaprezentowana dokładna postać V-monotonicznego standardo- wego rozkładu gaussowskiego (a konkretnie jego gęstości, gdyż jest to miara absolutnie ciągła względem miary Lebesgue’a na prostej). Wyjdziemy od funkcji tworzącej momenty, która została uzyskana w pracy „V-monotone independence”, a następnie skonstruuje- my analityczne rozszerzenie transformaty Cauchy’ego do górnej półpłaszczyzny. Ze wzoru odwrócenia Stieltjesa otrzymamy gęstość, przy okazji udowadniając bezatomowość miary.

Thursday, March 25, 2021 - 10:30, room zoom.us

Nahla Ben Salah (Sfax University and Carthage University-Tunisia)

Convolution of an infinitely divisible distribution and a Bernoulli distribution

In this paper, we essentially characterize the real power of the convolution set of $X + Y$ , where $X$ and$ Y \geq0$ are two independent random variables which have respectively a Bernoulli distribution, with parameter $r\in (0, 1)$, and an infinitely divisible probability distribution $\mu$. The above problem is equivalent to finding the set of $ x, y\geq0$ such that the mapping $z\mapsto (1 - r + re^ z )^{ x} (E(e^{ zY} )) ^{y}$ is a Laplace transform of some probability distribution. This class of real power of convolution x is provided and described.

Thursday, March 04, 2021 - 10:30, room zoom.us

Neil O'Connell

Some new perspectives on moments of random matrices

The study of 'moments' of random matrices (expectations of traces of powers of the matrix) is a rich and interesting subject, with fascinating connections to enumerative geometry, as discovered by Harer and Zagier in the 1980’s. I will give some background on this and then describe some recent work which offers some new perspectives (and new results). Based on joint work with Fabio Deelan Cunden, Francesco Mezzadri and Nick Simm.

Thursday, January 21, 2021 - 10:30, room zoom.us

Eugene Litvinov

From Stirling numbers to Stirling operators

We define and study a spatial (infinite-dimensional) counterpart of Stirling numbers. In classical combinatorics, the Pochhammer symbol $(m)_n$ can be extended from a natural number $m\in\mathbb N$ to the falling factorials $(z)_n=z(z-1)\dotsm (z-n+1)$ of an argument $z$ from $\mathbb F=\mathbb R\text{ or }\mathbb C$, and Stirling numbers of the first and second kinds are the coefficients of the expansions of $(z)_n$ through $z^k$, $k\leq n$ and vice versa. When taking into account spatial positions of elements in a locally compact Polish space $X$, we replace $\mathbb N$ by the space of configurations---discrete Radon measures $\gamma=\sum_i\delta_{x_i}$ on $X$, where $\delta_{x_i}$ is the Dirac measure with mass at $x_i$. The spatial falling factorials $(\gamma)_n:=\sum_{i_1}\sum_{i_2\ne i_1}\dotsm\sum_{i_n\ne i_1,\dots, i_n\ne i_{n-1}}\delta_{(x_{i_1},x_{i_2},\dots,x_{i_n})}$ can be naturally extended to mappings $M^{(1)}(X)\ni\omega\mapsto (\omega)_n\in M^{(n)}(X)$, where $M^{(n)}(X)$ denotes the space of $\mathbb F$-valued, symmetric (for $n\ge2$) Radon measures on $X^n$. There is a natural duality between $M^{(n)}(X)$ and the space $\mathcal {CF}^{(n)}(X)$ of $\mathbb F$-valued, symmetric continuous functions on $X^n$ with compact support. The Stirling operators of the first and second kind, $\mathbf{s}(n,k)$ and $\mathbf{S}(n,k)$, are linear operators, acting between spaces $\mathcal {CF}^{(n)}(X)$ and $\mathcal {CF}^{(k)}(X)$ such that their dual operators, acting from $M^{(k)}(X)$ into $M^{(n)}(X)$, satisfy $(\omega)_n=\sum_{k=1}^n\mathbf{s}(n,k)^*\omega^{\otimes k}$ and $\omega^{\otimes n}=\sum_{k=1}^n\mathbf{S}(n,k)^*(\omega)_k$, respectively. In the case where $X$ has only a single point, the Stirling operators can be identified with Stirling numbers. We derive combinatorial properties of the Stirling operators, present their connections with a generalization of the Poisson point process and with the Wick ordering under the canonical commutation relations. The talk is based on a joint paper with Dmitri Finkelshtein (Swansea), Yuri Kondratiev (Bielefeld) and Maria Joao Oliveira (Lisbon).

Tuesday 15th and Thursday 17th December 2020, room zoom.us

Polonium summary seminar

Two-days meeting summarizing the Franco-Polish project NAWA-PHC Polonium "Quantum structures and processes"

Thursday, December 3, 2020 - 10:45, room zoom.us

Lahcen Oussi

Analogue of Poisson type limit distribution in discrete bm-Fock space

The analogue of ”Poisson type” limit distribution is constructed on the discrete bm-Fock space, which is a kind of ”Fock space” generated by all the increasing sequence in some countable, partially ordered set associated with positive symmetric cones in Euclidean spaces, including $\mathbb{R}_{+}^{d}$, the Lorentz cone in Minkowski’s spacetime and the cone of positive definite real symmetric $(d\times d)$ matrices. The Analysis of the cones plays significant role in our study and the combinatorics is based on bm-ordered noncrossing partitions with blocks consisting of either one or two elements.

Thursday, 26. November 2020 - 10:45, room zoom.us

Adrian Dacko

V-monotoniczne centralne twierdzenie graniczne, cz. 3

W referacie przedstawimy transformatę Cauchy'ego V-monotonicznego rozkładu gaussowskiego. Korzystając ze wzoru odwrócenia Stieltjesa, wyznaczymy samą miarę i pochylimy się nad zagadnieniem istnienia atomów. Część absolutnie ciągłą uzyskamy w postaci uwikłanej.

Thursday, 19. November 2020 - 10:45, room zoom.us

Adrian Dacko

V-monotoniczne centralne twierdzenie graniczne, cz. 2

Thursday, 12. November 2020 - 10:30, room zoom.us

Wiktor Ejsmont

Cotangens

W referacie przedstawię nowe wyniki otrzymane z F. Lehnerem związane z funkcją trygonometryczną cotangens oraz ich związków z wolną probabilistyką oraz macierzami losowymi.

Thursday, 5. November 2020 - 10:45, room zoom.us

Mariusz Tobolski

Noncommutative universal spaces (continue)

In algebraic topology, by a universal space for a group G one usually means the Milnor universal space EG, which classifies paracompact principal bundles. This talk aims to propose an analog of the space EG in the context of locally compact \sigma-compact group actions on C*-algebras. Using this new notion, we define (locally trivial) noncommutative principal bundles with a locally compact \sigma-compact structure group. In the compact case, i.e. compact group actions on unital C*-algebras, this definition coincides with the finiteness of the local-triviality dimension introduced by myself, E. Gardella, P. M. Hajac, and J. Wu. Inspired by the work of N. C. Phillips on inverse limits of C*-algebras, I will work with the category of unital \sigma-C*-algebras with *-homomorphisms

Thursday, 29. October 2020 - 10:45, room zoom.us

Mariusz Tobolski

Noncommutative universal spaces

Thursday, 19. October 2020 - 10:45, room A

Yulia Kuznetsova

New norm estimates for functions of the Laplacian on the `ax+b' groups

Our aim is to find new estimates for the norms of certain functions of the (minus) Laplace operator $\cal L$ on the `$ax+b$' group. I will spend certain time presenting and comparing different methods. We estimate in particular norms of $\exp(-t\cal L^\gamma)$, $\gamma>0$, and of $(\cal L-z)^s$, with complex $z,s$, with asymptotics in $z$ and $s$. Next, and this is the central point, we estimate from below the $L^1$ and uniform norms of the kernels $k_t$ of wave propagators $\psi(\sqrt{\cal L})\exp(it \sqrt{\cal L})$, with $\psi\in C_0(\R)$. M\"uller and Thiele (Studia Math., 2007) have shown that $\|k_t\|_1\le C_\psi t$ and $\|k_t\|_\infty\le C_\psi$ as $t\to+\infty$. We prove that these estimates are sharp. This is joint work with Rauan Akylzhanov, Michael Ruzhansky and Haonan Zhang. The talk is a part of the Polonium programme, co-financed by the Polish National Agency for Academic Exchange (NAWA).

Thursday, 08. October 2020 - 10:45, room 602

Philip Cohen

Moments of discrete orthogonal polynomial ensembles

In joint work with Fabio Deelan Cunden and Neil O’Connell, building on earlier results by Ledoux (2005), we found hypergeometric representations for the factorial moments of the Charlier and Meixner orthogonal polynomial ensembles. In this talk, I will describe how, if the number of particles is suitably randomised, the factorial moments have a polynomial property, and satisfy three-term recurrence relations and differential equations. In particular, the normalised factorial moments of the randomised ensembles are precisely related to the moments of the corresponding equilibrium measures. I will also briefly outline how these results can be interpreted as Cauchy-type identities for certain Schur measures.

Thursday, 27. February 2020 - 10:15, room 602

Maciej Dołęga

Random Young diagrams and the approximate factorization property

We explain the concept of characters of the symmetric group with the approximate factorization property, introduced by Biane and developed by Śniady, and its role in the study of the asymptotic behavior of large Young diagrams. We discuss how to extend these ideas to study the asymptotic behavior of deformed Young diagrams, which arise from classical deformations of symmetric functions.

Thursday, 30. January 2020 - 10:15, room 602

Piotr Śniady

Poisson limit theorems related to random Young diagrams

We investigate a number of asymptotic questions related to Robinson-Schensted algorithm applied to a random input and show that the answer for each of them is given by the Poisson process. The first problem concerns the growth of the bottom rows of the Young diagram which is subject to Plancherel growth process. The second problem concerns the evolution in time of the position of the box with a specified number in the insertion tableau as new numbers are inserted into the tableau. The third problem concerns the shape of the bumping route when a specified number is inserted into a large Plancherel-distributed tableau and, in particular, the number of the row in which the bumping route enters a specified column

Thursday, 09. January 2020 - 10:15, room 602

Romuald Lenczewski

Niezależność ortogonalna

Postaram się przedstawić pojęcie niezależności ortogonalnej, która pojawia się w naturalny sposób w kontekście zasady subordynacji dla addytywnego i multiplikatywnego splotu wolnego, jak również w podejściu operadowym do niezależności nieprzemiennej. Posiada naturalny iloczyn grafów z korzeniem, który jest kanonicznie związany z iloczynem wolnym grafów. Omówię również inne pojęcia z tą niezależnością związane, takie jak iloczyn ortogonalny stanów, addytywny splot ortogonalny i ewentualnie multiplikatywny splot ortogonalny.

Thursday, 19. December 2019 - 10:15, room 602

Franz Lehner

The trace method for cotangent sums

The main result of this talk is a contribution to a popular problem from classical calculus of trigonometric function, namely the evaluation of sums integer powers of the cotangent in closed form and a combinatorial analysis of the coeffcients of the resulting polynomial expressions. These turn out to be positive integer valued polynomials with interesting combinatorial properties. Our main observation is that the cotangent values are the eigenvalue of a simple self-adjoint matrix and therefore the trace method is applicable.

Thursday, 07. November 2019 - 10:15, room 602

Adrian Dacko

V-monotoniczne centralne twierdzenie graniczne

W referacie przypomnimy pojęcie V-monotonicznej niezależności, sformułujemy odpowiednie centralne twierdzenie graniczne oraz omówimy jego kombinatorykę. Zostanie przedstawione podejście do uzyskania rekurencji na momenty parzystego rzędu standardowego V-monotonicznego rozkładu gaussowskiego (przy pomocy odpowiednich operatorów gaussowskich określonych na tzw. ciągłej V-monotonicznej przestrzeni Focka). Z tej rekurencji, rozwiązując odpowiednie zagadnienie początkowe, na które składa się równanie różniczkowe zwyczajne typu Abela (II-go rodzaju), otrzymuje się funkcję tworzącą momenty tego rozkładu w postaci uwikłanej oraz, co za tym idzie, jego część absolutnie ciągłą (także w postaci uwikłanej).

Thursday, 30. October 2019 - 10:15, room 602

Piotr M. Hajac (IMPAN)

V-monotoniczne centralne twierdzenie graniczneFrom pushouts to pullbacks: a sample of noncommutative topology

In topology, pushouts are formal recipes for collapsing and gluing topological spaces. For instance, shrinking the boundary circle of a disc to a point yields a sphere, shrinking the equator of a sphere to a point gives two spheres joined at the point, collapsing the boundary of a solid torus to a circle, or gluing two solid tori over their boundaries, produces a three-sphere. In noncommutative topology, such procedures are expressed in terms of pullbacks of C*-algebras. It turns out that one can visualize a pullback of C*-algebras of graphs as a pushout of these graphs thus providing much needed intuition to the abstract setting of operator algebras. The goal of this talk is to discuss how to make this visualization rigorous by conceptualizing abundant examples from noncommutative topology that lead to a new concept of morphisms of graphs. In particular, we replace the standard idea of mapping vertices to vertices and edges to edges by the more flexible idea of mapping finite paths to finite paths. (Based on joint works with Alexandru Chirvasitu, Sarah Reznikoff and Mariusz Tobolski.)

Thursday, 12. October 2019 - 10:15, room 602

Janusz Wysoczański

O pewnej C-algebrze generowanej przez częściowe izometrie i spektrum jej maksymalnej C-podalgebry abelowej

Operatory kreacji i anihilacji na słabo monotonicznej przestrzeni Focka są częściowymi izometriami. W referacie podam własności C*-algebry, którą generują. Opiszę także jej maksymalną abelową C*-podalgebrę (MASA). Dla tej MASA pokażę w jaki sposób można scharakteryzować jej przestrzeń Gelfanda czyli spektrum.

Thursday, 17. October 2019 - 10:15, room 602

Ryszard Szwarc

Closable Hankel forms and moment problems

In a paper from 2016 D. R. Yafaev considers Hankel operators associated with Hamburger moment sequences qn and claims that the corresponding Hankel form is closable if and only if the moment sequence tends to 0. The claim is not correct, since we prove closability for any indeterminate moment sequence but also for certain determinate moment sequences corresponding to measures with finite index of determinacy. It is also established that Yafaev’s result holds if the moments satisfy q1/(2n)2n = o(n).

Wednesday, 10.10.2019, 10:15-12:00, room 602

Marek Bożejko (IMPAN)

Positivity and Gaussian processes

We will present relations between positive definite kernels (also operator valued) and classical and quantum Gaussian processes following the paper of Yanqui Qiu - "On a result of Bozejko on extension of positive definite kernels" - arXiv - 2019 and Bull. London Math.Soc 2019 and the results of E.Hirai, T.Hirai and A.Hora and M.Guta and myself on Thoma representations of central positive definite functions on Coxeter groups of type A and B, i.e. on permutations group and sign-permutation group on infinite many letters.

Wednesday, 4.09.2019, 10:15-12:00, room 603
Vitonofrio Crismale (University of Bari Aldo Moro)
Sums and limit distribution for non symmetric weakly monotone position operators

In this talk we present the asymptotic vacuum distribution, under an appropriate scaling, of a family of partial sums of non-symmetric position operators on weakly monotone Fock space. This can be seen as the "Poisson type" limit measure in our setting. We preliminary show that any single operator has the vacuum law belonging to the free Meixner class. After establishing some relations between the combinatorics of Motzkin and Riordan paths, we give a recursive formula for the vacuum moments of the law of any partial sum. We exploit these results to achieve the asymptotic distribution as described above, which turns out to be indeed the sum of an atomic and an absolutely continuous part. Based on joint work with M.E. Griseta and J. Wysoczanski.

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