About the seminar

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

Organizers: prof. Marek Bożejko (the main organizer), Ilona Królak (technical organizer)

Time and place: usually every Wednesday, 12.15 - 14.00 in Institute of Mathematics, University of Wroclaw, room 603.

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, Artur Buchholz, Ilona Królak, Anna Krystek, Romuald Lenczewski (Politechnika Wrocławska), Wojciech Młotkowski, Rafał Sałapata (Politechnika Wrocławska), Piotr Śniady, Łukasz Wojakowski, Janusz Wysoczański

List of all seminars in the Institute of Mathematics, University of Wrocław

Current schedule:

Wednesday, 17 March, 2010, 12:30-14:00, room 603,

Alexander Bendikov

On a class of random walks on the group $S_\infty$

We will introduce a class of random walks on the infinite symmetric group for each of which we will be able to estimate (thanks to a very symple structure of the spectrum of the corresponding Laplacian) the transition function of the random walk under consideration.

13th Workshop: Non-Commutative Harmonic Analysis With Applications To Probability, Bedlewo 2009

Every year in Summer we organize a workshop in Bedlewo, Poland .

Quantum Probability and Applications to Complex Networks - lectures by Nobuaki Obata

Previous talks:

Wednesday, 10 March, 2010, 12:30-14:00, room 603,

Marek Bożejko

"Rozkłady nieskończenie podzielne w klasycznej i wolnej probabilistyce z zastosowaniami do problemu BMV"

Wednesday, 3 March, 2010, 12:15-14:00, room 603,

Marek Bożejko, Janusz Wysoczański, Anna Kula, Ilona Królak

Wrażenia z pobytu w Walii.

Wednesday, 20 January, 2010, 12:15-14:00, room 603,

Jan Florek

"On Barnette's conjecture".

Wednesday, 13 January, 2010, 12:15-14:00, room 603,

Romuald Lenczewski

"Asymptotic properties of random matrices and pseudomatrices".

We discuss the asymptotics of sums of matricially free random variables called random pseudomatrices and compare it with that of random matrices with block-identical variances. The concept of matricial freeness can be viewed as a `matricial version of freeness'. For both random matrices and random pseudomatrices we find the limit joint distributions of blocks and give their Hilbert space realizations, using operators called `matricially free Gaussian operators'. In particular, if the variance matrices are symmetric, the limit joint distributions of symmetric blocks are the same for objects of both types.

Wednesday, 6 January, 2010, 12:15-14:00, room 603,

Jarosław Zemanek (IMPAN Warszawa)

"On fixed points of holomorphic mappings".

In this joint work with Simeon Reich and David Shoikhet, we study the structure of the fixed point set of a holomorphic mapping defined on a (not necessarily bounded or convex) domain in a complex Banach space, by means of ergodic theory and a nonlinear numerical range.

Wednesday, 16 December, 2009, 12:15-14:00, room 603,

Wojciech Młotkowski

"Wolne i warunkowo wolne półgrupy miar z liniowymi współczynnikami Jacobiego".

Wednesday, 9 December, 2009, 12:15-14:00, room 603,

Rafał Sałapata

Próba wyznaczenia zależności między kumulantami monotonicznymi a addytywnym splotem monotonicznym miar z momentami

Wednesday, 4 November, 2009, 12:15-14:00, room 603,

Anna Kula i Janusz Wysoczański

Ruchy Browna na stożkach dodatnich związane z bm-niezależnoscią (część III)

Wednesday, 28 October, 2009, 12:15-14:00, room 603,

Anna Kula i Janusz Wysoczański

Ruchy Browna na stożkach dodatnich związane z bm-niezależnoscią (część II)

Wednesday, 21 October, 2009, 12:15-14:00, room 603,

Łukasz Wojakowski

Rozkłady nieskończenie podzielne w warunkowo wolnej probabilistyce

Wednesday, 14 October, 2009, 12:15-14:00, room 603,

Anna Kula i Janusz Wysoczański

Ruchy Browna na stożkach dodatnich związane z bm-niezależnoscią (część I)

Wednesday, 7 October, 2009, 12:15-14:00, room 603,

Piotr Śniady

Asymptotic representation theory of unitary groups and random matrix theory (joint work with Benoit Collins)

We show that representations of the unitary groups asymptotically behave like random matrices. For example: the decomposition of Kronecker tensor product of two irreducible representations behaves like a sum of two independent unitarily invariant random matrices with prescribed eigenvalues. Our main tool is to view representations as random matrices with quantum entries.

Wednesday, 20 May, 2009, 10:15-12:00, room 603,

Michał Wojciechowski (Polish Academy of Sciences)

Bounded Approximation Property of Sobolev spaces of functions with integrable gradient on arbitrary simply connected planar domain

Tuesday, 19 May, 2009, 10:15-12:00, room 601,

Rafał Latała (University of Warsaw)

Concentration inequalities for logarithmically concave measures

Monday, 18 May, 2009, 16:30-18:00, room 141, Institute of Computer Science Informatyki, ul. Joliot-Curie 15,

Jacek Wesołowski (Warsaw University of Technology)

Quadratic harnesses, q-commutation relations and Askey-Wilson polynomials

Kwadratowe harnessy to procesy stochastyczne mające liniową warunkową wartość oczekiwaną i kwadratową warunkową wariancję przy warunkowaniu przez przeszłość i przyszłość łącznie. Przykłady kwadratowych harnessów to: proces Wienera, proces Poissona, proces Dirichleta, wolny proces Wienera, procesy q-gaussowskie, proces bi-poissonowski. Problem, którym się zajmujemy, to próba kompletnego opisu tej klasy procesów stochastycznych. Okazuje się, że są to zawsze procesy markowskie, co więcej są one wyznaczone jednoznacznie przez 5 stałych rzeczywistych (jedną z nich jest q). Stałe te są parametrami równania q-komutacyjnego wiążącego dwie części macierzy Jacobiego ortogonalnych wielomianów martyngałowych rozważanego procesu. Macierz Jacobiego jest funkcją afiniczną czasu i wspomniane dwie części to odpowiednie współczynniki macierzowe tej właśnie funkcji. W taki schemat po odpowiednim przeparametryzowaniu zanurzyć można szeroką klasę wielomianów Askey-Wilsona. Miara ortogonalizująca te wielomiany jest rozkładem jednowymiarowym odpowiedniego kwadratowego harnessa. Wyniki, o których będę mówił, zostały otrzymane wspólnie z W. Brycem i częściowo z W. Matysiakiem.

Monday, 18 May, 2009, 14:30-16:00, room 141, Institute of Computer Science Informatyki, ul. Joliot-Curie 15,

Stanisław Lech Woronowicz (University of Warsaw)

Symmetries of Heisenberg double

Wednesday, 13 May, 2009, 10:15-12:00, room 603,

Romuald Lenczewski (Wrocław University of Technology)

Asymptotic properties of random matrices and pseudomatrices

I will discuss asymptotic properties of sums of matricially free random variables. The concept of matricial freeness can be viewed as a generalization of both freeness of Voiculescu and monotone independence of Muraki. The main idea is that we consider arrays of random variables. In particular, the `rows' of square arrays lead to free random variables whereas the `rows' of lower- (upper-) triangular arrays lead to monotone (anti-monotone) independent random variables. At the same time, the sums of all random variables in the given array, called a random pseudomatrix, have the same asymptotics as a Gaussian random matrix if we assume (in both cases) that the variance matrix is symmetric and has block-identical variances. The same feature is exhibited by the blocks of both objects.

Wednesday, 6 May, 2009, 10:15-12:00, room 603,

Benoit Collins (Université d'Ottawa)

Random matrix models arising from quantum information theory and applications

We will focus this talk on the study of a random matrix model describing the output of a specific input state (the Bell state) under well chosen random quantum channels. This matrix model is quite new from the point of view of random matrix theory and has many interesting properties. In particular, under appropriate scalings we can compute its asymptotic eigenvalue distribution and provide new examples and counterexamples for various additivity conjectures in quantum information theory.

Wednesday, 29 April, 2009, 10:15-12:00, room 603,

Karol Życzkowski (Jagiellonian University, Cracow, Poland and Center for Theoretical Physics, Polish Academy of Science)

Random density matrices and random quantum maps

A link between properties of quantized chaotic systems and random matrices will be reviewed. Statistical properties of periodically driven quantum chaotic systems described in a finite dimensional Hilbert space H_N can be described by circular ensembles of random unitary matrices. To describe the effect of a possible interaction of the system in question with an environment one needs to work with density operators, which are Hermitian, positive and normalized. Discrete time evolution of a density matrix can be represented by so-called quantum operation (completely positive, trace preserving map). We introduces several ensembles of random operations, discuss algorithms to generate them and investigate spectral properties of the corresponding superoperators with spectrum consisting of N^2 eigenvalues inside the unit disk. A quantum analogue of the Frobenius-Perron theorem concerning the spectrum of stochastic matrices is formulated. Obtained predictions for random operations are compared with spectral properties of quantized chaotic systems, interacting with an environment.

Wednesday, 22 April, 2009, 10:15-12:00, room 603,

Ilona Królak

Complex fermion hypercontractivity

I will present a new proof of the hypercontractivity inequalities for holomorphic algebras generated by elements fulfilling Canonial Anicommutation Relations (CAR).

Wednesday, 1 April, 2009, 10:15-12:00, room 603,

Tim Steger (Universita degli Studi di Sassari)

Free group representations from vector-valued multiplicative functions

Let \Gamma denote a noncommutative free group, and let \Omega stand for its boundary. We construct a large class of unitary representations of \Gamma. This class contains many previously studied representations, and is closed under several natural operations. Each of the constructed representations is in fact a representation of \Gamma\ltimes_\lambda C(\Omega). As representations of \Gamma\ltimes_\lambda C(\Omega), they are all irreducible. As representations of \Gamma, each of them is either irreducible, or the direct sum of exactly two irreducible, inequivalent \Gamma-representations.

Wednesday, 25 March, 2009, 10:15-12:00, room 603,

Troels Steenstrup Jensen (University of Southern Denmark)

Herz-Schur multipliers and spherical functions on the non–abelian free groups

There will be a short introduction to the basic concepts of the talk, namely spherical functions and Herz-Schur multipliers. The main focus of the talk is a criterion for determining when a radial function on a non-abelian free group is a Herz-Schur multiplier, together with an explicit formula for the Herz-Schur norm. This result, obtained by Haagerup-Szwarc, has remained unpublished but is to appear as part of a joint paper. As an application of this result one can find closed expressions for the Herz-Schur norm of the spherical functions on the non-abelian free groups, which extends previous work by Pytlik-Szwarc. Haagerup used these formulas to show that there are Herz-Schur multipliers on the non-abelian free groups which are not coefficients of uniformly bounded representations. This has remained unpublished, but an alternate proof has since been published by Pisier. Actually, a generalization can be found for countable discrete groups, using that such groups are amenable if and only if the Herz-Schur multipliers coincide with the Fourier-Stieltjes algebra (due to Bozejko) and that each Herz-Schur multiplier can be realized as the coefficient of a (not necessarily uniformly bounded) representation (due to Bozejko-Fendler). Finally, we will discuss various extensions to (some) Lie groups and general locally compact groups.

Wednesday, 18 March, 2009, 10:15-12:00, room 603,

Maciej Burnecki (Wrocław University of Technology)

Pewna operatorowa charakteryzacja przestrzeni L^{p}

Rozważamy zanurzenie grupy transformacji odwracalnych odcinka [0,1] w algebrę operatorów ograniczonych na przestrzeni Orlicza. Dowodzimy, że włożenie to zachowuje działanie grupowe wtedy i tylko wtedy, gdy przestrzeń Orlicza jest przestrzenią L^p dla pewnego 1\le p<\infty.

Wednesday, 11 March, 2009, 10:15-12:00, room 603,

Melanie Hinz

Multiplicative square of the free Poisson measure

We compute moments and free cumulants of the measure \rho_t:=\pi_t\boxtimes\pi_t, where \pi_t denotes the free Poisson law with parameter t>0. We also compute free cumulants of the symmetrization of \rho_t. Finally we introduce free symmetrization of a probability measure on \mathbb{R} and provide some examples.

Wednesday, 21 January, 2009, 10:15-12:00, room 603,

Paweł J. Szabłowski (Warsaw University of Technology)

On conditional q-normal distributions

We expand Chebishev polynomials and some of its linear combination in linear combinations of q-Hermite and Al Salam-Chihara polynomials. We use these expansions to expand q-Normal and related densities into infinite series of Chebishev polynomials and thus study probabilistic properties of these distributions including efficient simulation. The analysis includs also case q>1. Then distributions are discrete. In the case of conditional normal distribution the support consists of zeros of some Al-Salam-Chihara polynomilas. We find those zeros. We use this result to prove the existence of stationary random fields with linear regressions and thus close an open question posed by W. Bryc et al.. We prove this result by describing a discrete 1 dimensional conditional distribution. As a by product we generalize to q-series case, a well known formula (x+y)^{n}=\sum_{i=0}^{n} \binom{n}{k}i^{k}H_{n-k}\left( x\right) H_{k}\left( -iy\right) , where H_{k}\left( x\right) denotes k-th Hermite polynomial.

Wednesday, 14 January, 2009, 11:15-12:00, room 607,

Ana-Maria Stan (Universite de Franche Comte, Besancon)

The completely bounded multiplier norm of some Leinert sets in the free group

In this talk, we present a way to compute the completely bounded multiplier norm of the characteristic function of the set of generators in the free group

Wednesday, 14 January, 2009, 10:15-11:00, room 607,

Nizar Demni (Universit\'e Pierre et Marie Curie, Paris)

Ultraspherical type generating functions, generalized Stieltjes transforms and Markov transforms

I give the classification of families of orthogonal polynomials with ultraspherical type generating functions. This gives identities where a generalized Stieltjes transform of some probability distribution is expressed as the geometric mean of two Stieltjes transforms on the one side and as a powered Stieltjes transform of a probability distribution on the other side. Using Markov transforms, we get a family of symmetric probability distributions interpolating between the arcsine and the Wigner distributions. As a matter of fact, interesting, not yet solved, problems emerge: finding a convolution operation that interpolates between the r-convolution defined by M. Bozejko for r=1/2 and r=1, find a parallel to Young diagrams for which this probability distribution is the rescaled limiting shape.

Wednesday, 7 January, 2009, 10:15-12:00, room 603,

Marcin Marciniak (University of Gdańsk)

Some problems concerning extremal positive maps acting on type I factors

We will consider extremal elements of the cone of all positive maps acting between B(K) and B(H) where K and H are Hilbert spaces. We will show that every positive map with the property that rank \phi(P)\leq 1 for any one-dimensional projection P is a rank 1 preserver. It allows to characterize all decomposable extremal maps as those which satisfy the above condition. Further, we prove that every extremal positive map which is 2-positive turns out to be automatically completely positive. Finally we get the same conclusion for such extremal positive maps that rank \phi(P)\leq 1 for some one-dimensional projection P and satisfy the condition of local complete positivity. It allows us to give a negative answer for Robertson's problem in some special cases.

Wednesday, 17 December, 2008, 10:15-12:00, room 603,

Stefan Neuwirth (Université de Franche-Comté, Besancon)

Transfer between Fourier and Schur multipliers

We will review well-known techniques and a new theorem that permit to associate a Toeplitz Schur multiplier to a Fourier multiplier, and vice versa, and provide some applications.

Wednesday, 10 December, 2008, 10:15-12:00, room 603,

Aleksander Pełczyński (Polish Academy of Sciences)

Banach spaces of functions and measures orthogonal to a Sidon set

G-compact abelian group, \Gamma-its dual, S\subset\Gamma- an infinite Sidon set; \lambda-the normalized Haar measure on G, L_1(G)=L_1(G,\lambda), M(G)-the space of all regular Borel measures on G. We consider Banach spaces: L_1^{S^\perp}(G)=\{f\in L_1(G)\vert \int_G \gamma^{-1}(g)\lambda(dg)=0 \foral \gamma\in S\}, M^{S^\perp}(G)=\{\mu\in M(G) \vert \int_G\gamma^{-1}(g)\mu(dg) \forall \gamma\in S\}. Clearly L_1^{S^\perp}(G) is naturally isometric to a subspace of M^{S^\perp}(G). We investigate Banach space properties of these spaces. Answering a question of Bourgain [B] we show that L_1^{S^\perp}(G) and M^{S^\perp}(G) are not L_1-spaces, although they share some properties of the classical spaces. \begin{theorem} (i) L_1^{S^\perp}(G) (resp. M^{S^\perp}(G)} is not isomorphic to any Banach lattice. (ii) There are bounded non absolutely summing operators from L_1^{S^\perp}(G) (resp M^{S^\perp}(G) ) to a Hilbert space. (iii) If E is a separable subspace of M^{S^\perp}(G) which contains L_1^{S^\perp}(G) then E is uncomplemented in M^{S^\perp}(G). \end{theorem} (iii) answers in negative a question of Plichko and Yost who asked whether so called separable complementation property (=SCP) is hereditary. Note that M^{S^\perp}(G) is a subspace of M(G) which has SCP, i.e. every separable subspace of M(G) is contained in a separable complemented subspace of M(G). \begin{theorem} (j) If G is metrizable then L_1^{S^\perp}(G) has a basis. (jj) L_1^{S^\perp}(G) has the Dunford-Pettis property, i.e. every weakly compact operator from L_1^{S^\perp}(G) to an arbitrary Banach space takes weakly null sequences into norm null sequences. (jjj) L_1^{S^\perp}(G) and M^{S^\perp}(G) have the uniform approximation property =(UAP). \end{theorem} Recall that a Banach space X has UAP if \exists c\ge 1 and a function \Phi:N \to N such that for every finite-dimensional subspace F\subset X there is a finite rank operator T:X\to X such that ||T||\le c, T(f)=f for f\in F \dim T(X)\le \Phi(\dim F).

Wednesday, 3 December, 2008, 10:15-12:00, room 603,

Romuald Lenczewski (Wrocław University of Technology)

Matricially free random variables, part 2

I will discuss a new concept of noncommutative independence called matricial freeness, which lies somewhere in between freeness and random matrices. My talk will be similar to that in Bedlewo a few months ago, but the presentation may contain some more details.

Wednesday, 26 November, 2008, 10:15-12:00, room 603,

Romuald Lenczewski (Wrocław University of Technology)

Matricially free random variables

Wednesday, 19 November, 2008, 10:15-12:00, room 603,

Piotr Śniady

Explicit combinatorial interpretation of Kerov character polynomials as numbers of permutation factorizations

We find an explicit combinatorial interpretation of the coefficients of Kerov character polynomials which express the value of normalized irreducible characters of the symmetric groups S(n) in terms of free cumulants R_2,R_3,... of the corresponding Young diagram. Our interpretation is based on counting certain factorizations of a given permutation.

Wednesday, 29 October, 2008, 10:15-12:00, room 603,

Piotr Śniady

Explicit combinatorial interpretation of Kerov character polynomials as numbers of permutation factorizations

Wednesday, 22 October, 2008, 10:15-12:00, room 603,

Marek Bożejko

New examples of Generalized Brownian Motions and positive definite functions on Coxeter groups

We present some new classes of Generalized Brownian Motions for which the vacuum is trace state. The elementary model using random matrix was obtained by Bryc, Dembo and Jiang (Annals of Probab.) and next by another method was done by Artur Buchholz. The corresponding measure is the free product of classical Gaussian and the Wigner law or q-Gaussian and Wigner law. Connections with new classes of positive definite functions on Coxeter groups also will be done.

Wednesday, 8 October, 2008, 10:15-12:00, room 603,

Wojciech Młotkowski

Liczby Fussa-Catalana w nieprzemiennej probabilistyce

Używając metod z nieprzemiennej probabilistyki pokazujemy ze ciąg Fussa-Catalana \left({mp+r\atop m}\right)\frac{r}{mp+r} jest dodatnio określony o ile p\ge1, 0\le r\le p lub p\le0, p-1\le r\le0. Odpowiednią miarę probabilistyczną na \mathbb{R} oznaczamy przez \mu(p,r). Pokazujemy różne własności miar \mu(p,r), na przykład zależności \begin{eqnarray*} \mu(1+p,1)^{\boxtimes t}&=&\mu(1+tp,1),\\ \mu(p_1,r)\boxtimes\mu(1+p_2,1)&=&\mu(p_1+rp_2,r),\\ \mu(a,b)\vartriangleright\mu(a+r,r)&=&\mu(a+r,b+r), \end{eqnarray*} wyliczamy też momenty potegi boolowskiej \mu(p,1)^{\uplus t}. Pokazujemy też że jeśli 0\le 2r\le p, r+1\le p lub p\le 2r+1, p\le r\le 0 to \mu(p,r) jest \boxplus-nieskończenie podzielna.

Saturday, 16 August, 2008, 9:55-10:40, room 601,

Nizar Demni (Bielefeld Universitaet)

Intertwining in commutative and noncommutative probability

The first part exhibits some well known examples of intertwining of commutative Markov semigroups. The second part gives a group Theoretical point of view of interwining together with an illustration using the Heisenberg group.

Saturday, 16 August, 2008, 9:00-9:45, room 601,

Jonathan Novak (Queen's University, Kinston, Canada)

Two approaches to symmetric group characters: content polynomials and Kerov polynomials

Kerov introduced a class of polynomials which, when evaluated at free cumulants, yield (normalizations of) irreducible symmetric group characters. More recently, Corteel, Goupil, and Schaeffer have considered ćontent evaluation" of symmetric functions. For a distinguished class of symmetric functions, content evaluation also yields symmetric group characters. Thus there is an a priori connection between free cumulants and content evaluation. We will explore this connection. We also present connections with symmetric functions of Jucys-Murphy elements, and with the Laurent expansion of the unitary "Weingarten function" of Collins and Sniady. In particular we give an explicit expression for all coefficients in this Laurent expansion in terms of contents and characters.

Saturday, 16 August, 2008, 11:50-12:40, room 601,

P.K.Das (Indian Statistical Institute, Kolkuta)

Optimal Control on Two-level Quantum system with energy Cost Functional

The optimal control problem of the time evolution of quantum spin of Pauli two-level system subjected to an external field with the minimum energy function will be illustrated and formulated in terms of the quantum spin up and spin down states of the Pauli two-level system.

Saturday, 16 August, 2008, 11:00-11:45, room 601,

Dong Myung Chung (The Catholic University of Korea)

Stock-Price models and Option pricing

Financial mathematics may be regarded as the branch of the applied mathematics which are concerned with the financial markets. Generally, financial mathematics derives, and extends the mathematical models to describe the dynamics of stock prices in the financial market and uses stochastic calculus to obtain the fair price of the derivative of the stock. In terms of practice, financial mathematics also overlaps heavily with the field of financial engineering and computational finance. Arguably, all three are largely synonymous, although the latter two focus on applications, while the former focuses on modelling of stock prices and option pricing. In this talk, we will go through four subjects below which might have you an insight into the theory of stock price models and the theory of option pricing. We will first discuss some of stochastic processes which are suitable for an adequate description of the dynamics of underlying assets such as stocks. We will next discuss the concepts of arbitrage-free pricing and risk neutral valuation, which are indeed the fundamental concepts of the theory of option pricing. We finally introduce the MTS distributions and processes and use them to develop the GARCH option pricing model with the MTS innovations. The following is the structure of this talk : I. Stock-price models : 1. Binomial model : log returns can be modeled by a random walk - Cox-Ross- Rubinstein model 2. Diffusion models : log returns can be modeled by a Wiener process - Black- Scholes model 3. Pure jump models : log returns can be modeled by a Levy process - Levy models such as VG model , CGMY model II. Asset pricing models : 1. Absolute pricing models - Equilibrium pricing models - CAPM, 2. Relative pricing models - Arbitrage-free pricing - Black-Scholes model III. Option pricing: 1. No-arbitrage argument approach - Black-Scholes PDE 2. Risk-neutral valuation approach - Risk-neutral measure - Equivalent martingale measure IV. The MTS - GARCH option pricing model

Wednesday, 11 June, 2008, 10:15-12:00, room 603,

Nobuaki Obata (Tohoku University)

A limit theorem for a continuous time quantum walk on a growing graph

Let G=(V,E) be a connected, locally finite graph with a fixed origin o\in V, and A the adjacency matrix. We consider continuous-time quantum walks on G, which are classical stochastic processes defined by P(X(t)=x)=|\langle \delta_x, e^{itA}\delta_o\rangle|^2, x\in V, P(Y(t)=n)=\sum_{x\in V_n}|\langle \delta_x, e^{itA}\delta_o\rangle|^2, \quad n=0,1,2,\dots, respectively, where V=\bigcup_{n=0}^\infty V_n is the stratification. It is interesting to study asymptotic behavior of Y(t) when the graph G grows. We derive a limit theorem by means of the asymptotic spectral analysis of growing graphs with quantum probabilistic techniques.

Wednesday, 4 June, 2008, 11:15-12:00, room 602,

Alina Kargol (Maria Curie-Skłodowska University)

Methods of infinite dimensional analysis in the theory of Gibbs states

1. The existence of phase transitions in quantum crystals. A translation invariant system of interacting quantum anharmonic oscillators indexes by the elements of a simple cubic lattice Z^d is considered. For the translation and rotation invariant models on the lattice Z^d (d>=3) with the nearest neighbour interactions the existence of phase transitions has been proven. In the scalar case, for the translation invariant models with the nearest neighbour interactions on the lattice Z^d (d>=3) a weaker result - the existence of phase transitions of the first order (due to Landau classification) has been obtained. The proofs are based on the representation of local Gibbs states in terms of path measures and thereby on the use of the infrared estimates and the Garsia-Rodemich-Rumsey inequality. Keywords: phase transition, quantum anharmonic crystal, Euclidean approach. 2. Decay of correlations in quantum models. The decay of the two-point correlation function of N-component ferromagnetic quantum model on the lattice Z^d (with arbitrary d) is shown to be the same as the decay of the pair-interaction potential J_ll', if the temperature is above the transition point T_C(1) of the corresponding one-component model. Keywords: quantum model, decay of correlations.

Wednesday, 4 June, 2008, 10:15-11:00, room 602,

Dorota Kępa (Maria Curie-Skłodowska University)

Statistics of animals on quasi-bounded graphs

Statistics of vertex sets on quasi bounded graphs is considered. It is expressed by sums of weight functions over compatible collections of connected sets called animals. We present conditions that have to be satisfied to obtain estimates and prove convergence for unbounded, irregular graphs. The idea of the proof was initiated by L. R. Dobrushin and A. Bovier.

Wednesday, 14 May, 2008, 16:15-18:00, room WS,

Eugene Lytvynov (Swansea University)

Orthogonal polynomials for classical and free Levy processes

It is well known that, among all Levy processes, essentially only Brownian motion and Poisson process have a chaos decomposition property, which leads to a unitary isomorphism between the L^2 space of the process and the symmetric Fock space. In the case of a general Levy process, one may either use multiple stochastic integrals in orthogonalized power jump processes, or use an expansion in orthogonal polynomials of white noise. We will compare both approaches and derive a special class of Levy processes for which the corresponding infinite-dimensional polynomials have a nice structure. This class will be an infinite-dimensional counterpart of the Meixner class of orthogonal polynomials on the real line. Furthermore, we will discuss an extension of these results to the case of free Levy processes.

Wednesday, 14 May, 2008, 10:15-12:00, room 603,

Eugene Lytvynov (Swansea University)

Orthogonal polynomials for classical and free Levy processes

Wednesday, 7 May, 2008, 10:15-12:00, room 603,

Piotr Śniady

Derivatives of functions on the set of Young diagrams

We study derivatives of functions on the set of generalized Young diagrams with respect to various set of parameters (such as: shape of a Young diagram, free cumulants of a Young diagram, other kinds of cumulants). In this way we are able to find explicit formulas for characters expressed in terms of free cumulants (Kerov polynomials) and other kinds of cumulants. One of our main tools is the Stanley-F\'eray character formula.

Wednesday, 23 April, 2008, 10:15-12:00, room 603,

Jan Florek (Wrocław University of Economics)

Billard and the Diophantine approximation

Wednesday, 9 April, 2008, 14:15-16:00, room ???,

Piotr Śniady

Kerov character polynomials and mysterious dimensions of (co)homologies

Guest talk on Differential Geometry and Topology Seminar. Irreducible representations of the symmetric groups S_n are indexed by Young characters. There are many ways of parametrizing Young diagrams, but one of them surpasses all the other by its beauty: the one which uses "free cumulants" (also appearing in the random matrix theory and Voiculescu's free probability). Kerov character polynomials express the characters of the symmetric groups in terms of the free cumulants; they have a surprisingly rich structure and there are many open conjectures concerning them. During my talk I will concentrate on their conjectured connection with the dimensions of some (co)homologies of some mysterious objects (Schubert varieties?). The talk is intended to be non-technical.

Wednesday, 9 April, 2008, 10:15-12:00, room 603,

Romuald Lenczewski (Wrocław University of Technology)

Subordination in free probability

The talk will concern structures related to the subordination property for free additive and multiplicative convolutions. This includes the s-free product of Hilbert spaces, s-free convolutions, s-free independence and the s-free product of graphs.

Wednesday, 2 April, 2008, 10:15-12:00, room 603,

Janusz Wysoczański

Remarks on bm-independence

We shall discuss the notion of bm-independence and show the general construction of bm-product of algebras. Then we shall exhibit the related construction for graphs - bm-product of graphs - which generalizes the comb product and the markovian product of graphs. Next we will formulate the general form of the bm-Central limit Theorem associated with positive symmetric cones, and explain the proof of it. Finally, we shall present an example of the Donsker's invariance principle for bm-independence.

Wednesday, 26 March, 2008, 10:15-12:00, room 602,

Abdessatar Barhoumi (Higher School of Sciences and Technologies, Hammam-Sousse, Sousse, Tunisia)

On Negative Binomial White Noise. Quantum Fields

The main purpose of this talk is to derive a white noise calculus for the negative binomial process, studying in details an associated family of field operators. In particular, by using higher powers of negative binomial Jacobi fields, we give an identification of the square of white noise algebra and a representation of the Virasoro algebra.

Wednesday, 19 March, 2008, 10:15-12:00, room 603,

Wojciech Młotkowski

Remarks on positve-definite functions on Coxeter groups

Wednesday, 5 March, 2008, 8:15-10:00, room B,

Philippe Biane (Université de Marne-la-Vallée)

Pitman theorem and Brownian motion on matrices

We show how to extend the classical theorem of Pitman on Brownian motion to higher dimensions.

Monday, 3 March, 2008, 16:30-17:30, room B,

Philippe Biane (Université de Marne-la-Vallée)

Kerov polynomials

We introduce Kerov polynomials and describe some of their properties.

Monday, 3 March, 2008, 15:00-16:00, room B,

Stanisław Lech Woronowicz (University of Warsaw)

GNS-maps and their applications in the theory of locally compact quantum groups

In the case of weights (infinite states) GNS construction is a little more complicated. For normalised states we deal with map A\ni a\mapsto a\Omega\in H, where \Omega is a cyclic vector. For weights this mapping is replaced by so called GNS map that should be treated as unbounded densely defined closed operator acting from the algebra into Hilbert space. The application of this concept to Tomita-Takesaki theory and to locally compact quantum groups will be discussed.

Monday, 3 March, 2008, 14:00-14:50, room B,

Andrzej Żuk (Université Paris 7)

Grupy automatów

Wednesday, 27 February, 2008, 10:30-12:00, room 607,

Marek Bożejko

Levy processes of Meixner type in noncommutative probability

We present a class of stationary quantum stochastic processes with (free, conditionally free, monotonically independent) increments, whose conditional moments satisfy of Laha-Lukacs formulas. They all have classical version Markov processes. The results are obtained together with W. Bryc and N. Demni.

Wednesday, 30 January, 2008, 10:15-12:00, room 607,

Anna Kula (Jagiellonian University in Kraków)

(p,q)-convolution and limit theorem for q-normal, (p,q)-commuting operators

In 2000 Carnovale and Koornwinder [G. Carnovale, T.H. Koornwinder, A q-analogue of convolution on the line, Methods Appl. Anal. 7 (2000), 705-726] defined a q-convolution. They proved that for some classes of measures it is associative and commutative. No positivity-preserving properties were disscuced. That question was posed by M. Bozejko on the conference in Bedlewo 2007. We anwser it partially. The notions of q-positivity and q-moments were studied in [A. Kula, A q-analogue of complete monotonicity, Colloq. Math. 111 (2008), 169-181]. In the talk we start from the algebraic interpretation of q-positivity and this leads us to a definition of (p,q)-convolution. It has a form similar to the q-convolution of Carnovale and Koornwinder. As for the properties, it is associative, commutative and symmetric with p and p^{-1}. Moreover, a (kind of) positivity-preserving property follows immediately form the algebraic construction. For the new convolution we find an appropriate analogue of Fourier transform and also present a central limt theorem. The above is a joint work with Eric Ricard (UFR, Besancon).

Wednesday, 23 January, 2008, 10:15-12:00, room 607,

Marek Bożejko

Non-commutative Riesz products on some discrete groups with applications to non-commutative Khinchine inequality and completely bounded multipliers

We present the contruction of posiive definite functions on some class of discrete groups (Coxeter, free product groups), which is clasical Abelian groups is Riesz product function. We also present applications to non-commutative Khinchine inequality and Boolean and Coxeter probability . Connections with completely bounded multipliers (Schur multipiers) on groups also will done.

Wednesday, 16 January, 2008, 10:15-12:00, room 607,

Zbigniew Palmowski

Queues and Young Tableaux

The talk gives short survey of some recent results connecting random matrices, non-colliding processes and queues in series. In particular, our main goal will be to prove the following result discovered by Baryshinkov (2001) and Gravner, Tracy and Widom (2001). Let B(t)=(B_1(t),..., B_n(t)) be a standard n-dimensional Brownian motion. Then the random variable: M_n=sup_{0 < s_1 < ... < s_{n-1} < 1} sum_{i=0}^{n-1} (B_i(s_{i+1})-B_i(s_i)) has the same law as the largest eigenvalue of an n-dimensional GUE random matrix. What is even more surprising another approach is through a celebrated Burke theorem, which will be also given. Finally, I hope to mention few very new results and related open problems.

Wednesday, 9 January, 2008, 10:15-12:00, room 607,

Nizar Demni (Universit\'e Pierre et Marie Curie, Paris)

Topics on matrix-valued stochastic processes

We exhibit some properties of matrix-valued stochastic processes and their corresponding eigenvalues. A deep insight to the latter makes use of the so-called radial Dunkl processes with classical reduced root systems. This allows to recover and answer probabilistic questions using determinantal representation of some special functions on some complex Lie algebras. The last part will be concerned with construction via matrix theory of some free processes.

Wednesday, 12 December, 2007, 10:15-12:00, room 607,

Artur Buchholz

The copulas in R^n

The discrete method for constructing the copulas in R^n will be presented. Next the non-discrete meaning for the previous "deus ex machina" will be dredged up.

Wednesday, 21 November, 2007, 10:15-12:00, room 607,

Artur Buchholz

Positive definite functions on 2-partitions and representations of S_{\infty}

We show the method for constructing positive definite functions on 2-partitions using positive defined functions on S_{\infty}.

Wednesday, 14 November, 2007, 10:15-12:00, room 607,

Ilona Królak

Complex hypercontractivity

Wednesday, 24 October, 2007, 10:15-12:00, room 607,

Artur Buchholz

Positive defined functions on 2-partitions and representations of S_{\infty}

We show the method for constructing positive defined functions on 2-partitions using positive defined functions on S_{\infty}.

Friday, 19 October, 2007, 10:15-12:00, room 607,

Piotr H. Hajac (Polish Academy of Sciences, Warsaw)

Noncommutative join construction

The aim of this talk is to show how to carry out the join construction of compact quantum groups avoiding braiding and replacing the unit interval by an arbitrary unital C*-algebra (noncommutative compact Hausdorff space). This is done in terms of equivariantly projective Hopf-Galois extensions of C*-algebras. The completion of the extended algebra is a natural candidate for a non-crossed product example of a principal extension of C*-algebras in the sense of Ellwood (non-trivial noncommutative principal bundle). The main point is a general and explicit formula for a strong connection, which puts us directly into the framework of the index pairing between K-theory and K-homology. (Based on a joint work with L. Dąbrowski and T. Hadfield.)

Wednesday, 17 October, 2007, 10:15-12:00, room 607,

Yuriy Kryakin

On constants in Whitney's theorem and Jackson-Stechkin theorem

This is an introductory talk on history and current state of the theory of approximation.

Wednesday, 10 October, 2007, 10:15-12:00, room 607,

Wojciech Młotkowski

Combinatorial relation between free and conditionally free cumulants.

Wednesday, 3 October, 2007, 10:15-12:00, room 607,

Rafał Sałapata (Wrocław University of Technology)

Combinatorics related to two-parameter interpolation between free and monotone gaussian operators.

Past schedule for the year 2004/2005
Past schedule for the year 2005/2006
Past schedule for the year 2006/2007