# Discrete harmonic analysis seminar - past schedule for the year 2005/2006:

Current page of the seminar is available here.
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 Boďż˝ejko (the main organizer), Piotr ďż˝niady (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,....
 June 14, 2006, Wednesday, 10.15-12.00, room 607: Benoit Collins: Integration over compact quantum groups Compact quantum groups have been introduced by Woronowicz in the late eighties and one of their main feature is the existence of a Haar measure. The problem of computing the Haar measure on characters was solved by Woronowicz himself. On the other hand, computing the Haar measure on general functions on the quantum groups was left open. We propose here a method for computing it. In particular we focus on results and applications in the case of free compact quantum groups of Wang. June 7, 2006, Wednesday, 10.15-12.00, room 607: Serban Belinschi: Time behaviour for convolution semigroups "A borel probability measure $\mu$ that is infinitely divisible with respect to a given convolution usually belongs to a convolution semigroup $\{\mu_t\colon t\geq 0,\mu_1=\mu\}.$ In this talk I plan to address some aspects of the behaviour of the convolution of $\mu_t$ with another (not necessarily infinitely divisible) probability measure $\nu$, for free additive and free multiplicative convolutions, as a function of $t$, for $t$ near zero. In particular, we will show that the absolutely continuous part of $\nu$ is the a.e.-limit as $t$ tends to zero of the absolutely continuous part of the convolution of $\mu_t$ with $\nu$. The same methods will allow us to understand the time behaviour near one for the partial free convolution semigroups defined by Nica and Speicher and their multiplicative analogue. If time permits, we will also cover some aspects of the $t$-transform of the free additive convolution semigroup of Nica and Speicher comment on possible similar results for c-free convolution. Part of this is joint work with Alice Guionnet and part with Andu Nica." May 31, 2006, Wednesday, 10.15-12.00, room 607: Matthew Szczesny: Vertex algebras and mathematical approach to conformal field theory During the talk I will present the definition of a vertex algebra (introduced by Borcherds in 1986) and some examples. I will also sketch the construction of some bundles called "conformal blocks" over moduli spaces of algebraic curves. May 17, 2006, Wednesday, 10.15-12.00, room 607: Rafaďż˝ Saďż˝apata: Spectral properties and cyclic decomposition of the free product of graphs - examples We define the free product of graphs and decompose its adjacency matrix as a sum of free copies of adjacency matrices of initial graphs. In few examples of such a products we compute the full spectrum of the adjacency matrix by finding the cyclic decomposition. May 10, 2006, Wednesday, 10.15-12.00, room 607: Viatcheslav Belavkin (Nottingham): On Boson-Fermion q-monotone interpolation and quantum q-martingales Building on the nonadapted monotone quantum stochastic calculus introduced in [1], we give an explicit nonadapted representation of quantum Azema  q-martingales interpolating between the Boson (q=+1) and Fermion (q=-1) vacuum Gaussian states. They are simply defined as "q-monotone" quantum stochastic integrals of the vacuum white noise, and correspond to the "free monotone" Brownian motion in the vacuum-adapted case q=0. If time allows, we will also discuss the general quantum Ito-algebra q-monotone deformations of the tensorial quantum Levy processes, including the noncommutative temperature Wiener and Poisson type three states. References: [1] V. P. Belavkin: A Quantum Nonadapted Ito Formula and Stochastic Analysis in Fock Scale.  J of Funct Analysis 102 (2), p. 414--447 (1991). arXiv:math-ph/0512076 April 26, 2006, Wednesday, 10.15-12.00, room 607:  Janusz Wysoczanski: Combinatorics related to non-commutative bm-central limit theorems on symmetric cones Non-commutative central limit theorems for some symmetric cones, related to bm-independent families of random variables,  provide limit measures which are described in terms of recurence relation for moments. We shall show that the same can be obtained by considering some combinatorial objects associated with the symmetric cones, independently of any non-commutative probability. These objects consist of sequences of elements of the cone, associated with boolean-monotonic non-crossing pair partitions. We shall show a recurence relations for the cardinalities of these objects, which approximates the recurences for the bm-CLT. April 20, 2006, Thursday, 10.15-12.00, room EM: Piotr Hajac : Non-crossed-product examples of principal extensions of C*-algebras The aim of this talk is to present a method for proving that a given C*-algebra equipped with a free action of U(1) is not a crossed product  with its fixed-point subalgebra. It is based on the standard Fourier analysis with coefficients in a C*-algebra, and the computation of an appropriate K-invariant (Fredholm index) of a finitely generated projective module associated to the U(1)-action. This is in analogy with the classical fact that, if a principal bundle admits an associated vector bundle which is not trivial, then it itself cannot be trivial. (Based on the joint work with R.Matthes and W.Szymanski "Noncommutative index theory  for mirror quantum spheres", available at PMH's home page.) April 19, 2006, Wednesday, 10.15-12.00, room 607: Melanie Hinz : Solutions of QDE's of Hudson-Parthasarathy type Let h be the Hilbert space of the observed system and let H be the same for the heat bath. Consider the tensor algebra M generated by pairs (u,v) with u,v\in h^{\otimes n} and unitary evolutions \{ U_{s,t}\}_{0\leq s\leq t< \infty} in h\otimes H satisfying certain axioms, in particular those of stationarity and of independent increment. The aim is to show that there exist (i) a *-representation of M in a seperate Hilbert space k_0, (ii) operator L\in B(h,h\otimes k_0), self-adjoint operator H\in B(h), W unitary in h\otimes k_0 such that the unique unitary solution \widehat{U}(t) of the Hudson-Parthasarathy equation in h\otimes \Gamma(L^2(\R_+,k_0)): d\widehat{U}(t)=( \sum_{j} L_{j}^{*} dA_j-L_j dA_j^+ +\sum_{j,k}(W_k^j-\delta_k^j)d\Lambda_j^k +\left(-{\frac{1}{2}}\sum\limits_j L_j^*L_j+iH  \right)dt) \widehat{U}(t) is unitarily isomorphic with the given family U_{0,t}. April 12, 2006, Wednesday, 16.15-18.00, room 603: Quanhua Xu (Besancon): Applications of Quantum Probability to Operator Spaces April 12, 2006, Wednesday, 10.15-12.00, room 607: Adam Osďż˝kowski : Inequalities for weakly dominated martingales Odczyt bedzie poswiecony badaniu wlasnosci martyngalow (M_n), (N_n) adaptowanych do tej samej filtracji (F_n), takich, ze (M_n) jest slabo dominowany przez (N_n). Pojecie slabej dominacji oznacza, iz dla dowolnego n i dowolnej niemalejacej funkcji wypuklej f,            Ef(|dM_n||F_{n-1}) < Ef(|dN_n||\F_{n-1}).   W szczegolnosci, beda nas interesowac nierownosci: 1. Slabego typu: istnieje stala C taka, ze dla dowolnych t>0, n zachodzi t P(|M_n|>t) < \E |N_n|,  2. Silnego typu: dla 1 d/4 suffices to imply that f is bounded. In the case we are interested in, the desired condition involves the whole sequence |A^{n}f||^{1/n} and depends on behavior of the ultracontractivity function m. April 4, 2006, Tuesday, 14.15-16.00, room 710: M. Popa Combinatorial Hopf Algebras and Non-Commutative Polynomials About two decades ago, G-C Rota described the (Reduced) Incidence Hopf Algebras associated to partially ordered sets. One can find remarkable examples that fall in this category, such as the Connes-Kreimer Hopf Algebras of rooted trees. New interesting examples are inspired by the recent work on tree Hopf algebras in Quantum Electrodynamics and by some aspect of Free Probability. March 29,  2006, Wednesday, 10.15-12.00, room 607: NO SEMINAR. March 23, 2006, Thursday, 10.15-12.00, room ???: Javier Parcet (Madryd) Mixed norms of free variables We shall present a variation of the free analogue of Rosenthal inequality, recently obtained by Junge, Xu and the speaker. This result provides a general method to construct complete embeddings between noncommutative $L_p$ spaces, equipped with their natural operator space structures. In particular, given $1\leq p < q \leq 2$, we shall see that $\ell_q$ cb-embeds into $L_p(\mathcal{A})$ for some hyperfinite type III factor $\mathcal{A}$. This generalizes Junge's theorem on the embedding of OH into the predual of a von Neumann algebra and some related results by Pisier and Xu. Joint work with Marius Junge. March 22, 2006, Wednesday, 10.15-12.00, room 607: Marek Cwiklowski "2-dimensional generalization of the interactive Fock space and quantum decomposition" The interactive Fock space costructed with  3 operators: creator, anihilator and diagonal operator determines uniquely the probabilistic measure defined on the real line. The fundamental link between the sum of theese operators and the operator of the multiplication by x defined on the algebra generated by the family of orthogonal polynomials (so called quantum decompostion) is well known. Is possible to generalize the interactive Fock space in such a way, to obtain the planar measure and the family of orthogonal polynomials of 2 variable? I think, the answer is positive. March 15,  2006, Wednesday, 10.15-12.00, room 607: Krzysztof Stempak Transformaty Riesza dla rozwiniďż˝ďż˝ ortogonalnych (teoria $L^2$). Odczyt bazuje na wspolnej pracy z A.Nowakiem, L^2 theory of Riesz transforms for orthogonal expansions'', w ktorej zaproponowana zostala teoria transformat Riesza dla rozwiniec ortogonalnych w wielowymiarowym ujeciu. Teoria poparta jest licznymi przykladami dotyczacymi m.in. klasycznych rozwiniec. Omowiony jest takze przypadek rozwiniec stowarzyszonych z regularnym badz singularnym zagadnieniem Sturma-Liouville'a. March 8,  2006, Wednesday, 10.15-12.00, room 607: Adam Majewski O strukturze odwzorowan dodatnich Zostanie podana charakteryzacja odwzorowan dodatnich rozkladalnych.  Uwagi charakteryzujace strukture odwzorowan dodatnich zostana przedstawione. Wyklad bedzie bazowal na teorii algebr Jordana. March 1,  2006, Wednesday, 10.15-12.00, room 607: Marek Bozejko Bessis-Moussa-Villani conjecture and generalized Gaussian random variables The BMV conjecture for traces, which states that the function x -> Tr exp(A+i x B) is the Fourier transform of a positive measure on the real line, is true for A, B which are generalized Gaussian random variables. The case when A, B were free semicircle (free Gaussian) was proved by M. Fanned and D. Petz. February 23 lutego, godz. 10.15-12.00, room EM Uwe Franz (Universite de Franche-Comte, Besancon, Francja) "On infinitely divisible probability measures for the monotone convolution" (joint work with O.E. Barndorf-Nielsen) We recall Muraki's definition of the monotone convolution for probability measures and his Levy-Khinchine formula. Then we define and study stable measures for this convolution. Assuming embeddability and uniqueness of embedding one obtains a classification of monotonically stable measures analogous to the classical case. In particular, the Cauchy distribution is again stable. February 22,  2006, Wednesday, 10.15-12.00, room 607 ďż˝ukasz Wojakowski "On Bargmann representations of t-deformed measures " We will start by recalling the definitions concerning the t-deformation of measures and of the associated convolutions and limiting measures. We also discuss the concept of Bargmann representations of probability measures on the real line, which are probability measures on the complex plane which preserve some orthogonality properties. We discuss the behaviour of Bargmann representations under the t-deformation and calculate it explicitly for several important examples arising as limiting measures. January 25,  2006, Wednesday, 10.15-12.00, room 607 Attila Andai "On some open problems in information geometry" The seminar is about some conjectures, which are related to differential geometry, majorization relation and means. In quantum information geometry the scalar curvature at every state measures the average statistical uncertainty of the state. This is one of the basic idea of Petz's Conjecture, which is about the monotonicity of the scalar curvature with respect to the majorization relation when the state space is endowed with the Kubo--Mori metric. Cencov introduced the $\alpha$-connections and $\alpha$-geometry on the space of classical probability distributions and it was developed by Amari. Gibilisco and Isola showed that the idea of Petz's Conjecture can be extended to $\alpha$-geometries, they have another Conjecture about the monotonicity of the scalar curvature when the classical and the noncommutative probability space is endowed with $\alpha$-geometry. These Conjectures can written as simple inequalities, which are related to different means (for example Petz's Conjecture is related to the logarithmic mean). January 11,  2006, Wednesday, 10.15-12.00, room 607 Marek Bozejko "Free Levy-Meixner processes and random Wishart matrices" The free Meixner laws arise as the distributions of orthogonal polynomials with constant-coefficient recursions. We show that these are the laws of the pairs of random variables which have linear regressions and quadratic variationces when conditioned with respect to their sum. We also describe free Levy processes with quadratic conditional variances and  to prove a converes implication related to asymptotic freenes of random Wishart matrices- results of M.Capitaine and M.Casalis Indiana Univ.Math.J.53(2),397-431,2004. This is paper with Wlodzimierz Bryc. January 4,  2006, Wednesday, 10.15-12.00, room 607 Anatol Odzijewicz (Uniwersytet w Bialymstoku):  "C*-algebry z polaryzacja i odwzorowanie stanow koherentnych" ("C*-algebras with polarisation and maps of coherent states") Waďż˝nďż˝ rolďż˝ w kwantowym opisie ukďż˝adďż˝w fizycznych odgrywajďż˝ C*-algebry, ktďż˝re posiadajďż˝ strukturďż˝ podobnďż˝ do struktury zespolonej rozmaitoďż˝ci Kaehlerowskiej. W moim referacie omďż˝wiďż˝ zwiďż˝zek tych algebr z kwantyzacjďż˝ geometrycznďż˝ Kostanta-Souriau. Pokaďż˝ďż˝ rďż˝wnieďż˝, jak moďż˝na, startujďż˝c z takiej C*-algebry, zrekonstruowaďż˝ odwzorowanie stanďż˝w koherentnych. Podam kilka przykďż˝adďż˝w (kwantowa przestrzeďż˝ Minkowskiego, rďż˝ďż˝ne wersje zdeformowanego oscylatora harmonicznego) zastosowaďż˝ tego typu struktur. Bibliografia: A. Odzijewicz, Non-commutative Kahler structures, http://arxiv.org/abs/math-ph/0505061 December 21,  2005, Wednesday, 10.15-12.00, room 607: Anna Krystek:  "Miary nieskonczenie podzielne dla splotu warunkowo wolnego" December 14,  2005, Wednesday, 10.15-12.00, room 607: Marcin Marciniak:  "Rozkďż˝adalnoďż˝ďż˝ odwzorowaďż˝ dodatnich na C*-algebrach" ("Decomposability of positive maps on C*-algebras") Abstrakt: W trakcie referatu zamierzam przypomnieďż˝ podstawowe definicje i fakty dotyczďż˝ce pojďż˝cia rozkďż˝adalnoďż˝ci dodatniego odwzorowania liniowego $T:A\to B(H)$, gdzie $A$ jest C*-algebrďż˝, zaďż˝ $H$ pewnďż˝ przestrzeniďż˝ Hilberta. Nastďż˝pnie przedyskutujemy trzy przypadki: 1) $A$ i $B(H)$ sďż˝ algebrami macierzy kwadratowych o wspďż˝ďż˝czynnikach zespolonych. Przytoczďż˝ twierdzenie Stormera-Woronowicza o tym, ďż˝e w przypadku, gdy $A=M_2$ i $B(H)=M_n$, $n=2,3$, kaďż˝de odwzorowanie dodatnie jest rozkďż˝adalne i opiszďż˝ nowy dowďż˝d tego twierdzenia. Nastďż˝pnie pokaďż˝e, ďż˝e metoda zastosowana w tym dowodzie moďż˝e byďż˝ stosowana do analizy przypadku, gdy $n$ jest dowolnďż˝ liczbďż˝ naturalnďż˝ wiďż˝kszďż˝ od 1. 2) $T:A\to A$ jest dodatnim rzutowaniem zachowujďż˝cym jedynkďż˝. Uzasadnimy, ďż˝e w tym przypadku wďż˝asnoďż˝ďż˝ rozkďż˝adalnoďż˝ci ma istotny zwiďż˝zek ze strukturďż˝ algebry Jordana na obrazie $T(A_h)$ samosprzďż˝ďż˝onej czeďż˝ci algebry $A$. Nastďż˝pnie korzystajďż˝c z wynikďż˝w Stormera, Effrosa i Robertsona podamy konstrukcjďż˝ nierozkďż˝adalnego odwzorowania dodatniego z $M_4$ w $M_4$. Jeďż˝li czas pozwoli przedyskutujemy przypadek 3) $A=C^*(F_2)$ jest C*-algebrďż˝ zredukowanďż˝ grupy wolnej. Pokaďż˝emy, ďż˝e warunkiem koniecznym rozkďż˝adalnoďż˝ci odwzorowania dodatniego jest jego rozszerzalnoďż˝ďż˝ (wďż˝asnoďż˝ďż˝ tďż˝ rowaďż˝aďż˝ Arveson w kontekďż˝cie odwzorowaďż˝ caďż˝kowicie dodatnich). Nastďż˝pnie podamy konstrukcjďż˝ dodatniego odwzorowania nierozszerzalnego na algebrze grupowej grupy wolnej. December 7,  2005, Wednesday, 10.15-12.00, room 607: Romuald Lenczewski:  "Produkt wolny grafow" ("Free product of graphs") November 30,  2005, Wednesday, 10.15-12.00, room 607: Taku Matsui (Kyushu University, Fukuoka, Japan):  "Bose-Einstein Condensation on Graph" Abstract: We consider a mathematical model of Bose particles hopping on general graphs. We consider condition of occurence of Bose-Einstein Condensation. Mathematical ingredients are (a) the density of states for discrete Laplacians and adjacency matirices for infinite graphs. (b) transiency of the simple random walk on infinite graphs. November 23,  2005, Wednesday, 10.15-12.00, room 607: Piotr Sniady:  "Asymptotyczna teoria reprezentacji grup i algebr Liego" ("Asymptotic theory of representations of Lie groups and algebras") Streszczenie. Interesuja mnie dwa zagadnienia: (1) dla ustalonej algebry/grupy Liego rozwazam ciag reprezentacji (przywiedlnych lub nieprzywiedlnych), ktory w pewnym sensie dazy do nieskonczonosci; (2) rozwazam jedna z klasycznych serii algebr/grup Liego i dla kazdej algebry/grupy rozwazam pewna reprezentacje. Co mozna powiedziec o asymptotycznym zachowaniu takich ciagow reprezentacji? W moim odczycie pokaze, ze takie reprezentacje mozna traktowac jak macierze losowe o niekomutujacych wyrazach, ktore asymptotycznie zachowuja sie jak zwykle macierze losowe. November 16,  2005, Wednesday, 10.15-12.00, room 607: Piotr Pragacz (Polish Academy of Sciences): "Przechadzka po ogrodzie funkcji symetrycznych" ("A walk in the garden of symmetric functions") November 9,  2005, Wednesday, 10.15-12.00, room 607: Rďż˝mi Lďż˝andre (Universitďż˝ de Bourgogne): "Malliavin Calculus of Bismut type without probability". Abstract: In the first part, we do the translation of Malliavin Calculus in semi-group theory, by evacuating the probabilistic language. In the second part, I do the translation in semi-group theory of the lower-bound of Varadhan estimates I got a long time ago by Malliavin Calculus. November 2,  2005, Wednesday, 10.15-12.00, room 607: Artur Buchholz (Uniwersytet Wrocďż˝awski): "Zdeformowane macierze Pauliego i nieprzemienny ruch Browna 2" October 26,  2005, Wednesday, 10.15-12.00, room 607: Artur Jez (Uniwersytet Wrocďż˝awski):  "Combinatorial calculus 2" October 19,  2005, Wednesday, 10.15-12.00, room 607: Artur Jez (Uniwersytet Wrocďż˝awski):  "Combinatorial calculus 1 October 12,  2005, Wednesday, 10.15-12.00, room 607: Janusz Wysoczanski (Uniwersytet Wrocďż˝awski): "Centralne twierdzenia graniczne w probabilistyce monotonicznej zwiazanej ze zbiorami czesciowo uporzadkowanymi 2" October 5,  2005, Wednesday, 10.15-12.00, room 607: Janusz Wysoczanski (Uniwersytet Wrocďż˝awski): "Centralne twierdzenia graniczne w probabilistyce monotonicznej zwiazanej ze zbiorami czesciowo uporzadkowanymi 1" September 28,  2005, Wednesday, 10.15-12.00, room 607: Artur Buchholz (Uniwersytet Wrocďż˝awski): "Zdeformowane macierze Pauliego i nieprzemienny ruch Browna 1"