Discrete harmonic analysis seminar - past schedule for the year 2010/2011:
Current page of the seminar is available here.
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Discrete harmonic analysis seminar - past schedule for the year 2010/2011:Current page of the seminar is available here. |
I will describe a family of quantum groups which generalise at the same time some `easy' quantum groups of Teo Banica and Roland Speicher and the quantum isometry groups of the duals of the free groups introduced by Bhowmick and AS. The talk will be a continuation of the one given in April, the main results to be presented can be found in T. Banica and AS, `Two-parameter families of quantum symmetry groups', Journal of Functional Analysis 260 (2011), no.11, 3252-3282
The generator of the (semigroup of transition operators of the) Brownian motion is the Laplacian, and the generator of a general Levy process is a higher-order pseudo-differential operator. I will show that for a free Levy process (with finite variance), the generators are all singular integral operators, which will be written down explicitly. Moreover, in the centered case these generators are precisely the same as for an appropriately shifted free Brownian motion, a phenomenon with no classical analog.
Wystąpienie ma na celu kontynuacje kolejnych kroków dowodu zmierzającego do pewnych nowych charakteryzacji wolnych rozkładów Meixnera poprzez warunkową warjancje. W tym celu zostanie pokazana rekursja łącząca momenty splotu wolnych rozkładów Meixnera z pewnymi ciągami wyznaczającymi warunkowy drugi moment.
We tensor the C*-algebra of the equatorial Podle? quantum sphere with the algebra of continuous functions on the unit circle, act on the tensor product with the diagonal antipodal Z/2-action, and consider the invariant subalgebra. This gives a U(1)-C*-algebra A with the quantum real projective space C*-algebra C(RPq(2)) as its U(1)-invariant part. Using the identity representation of U(1), we associate with it a finitely generated projective module over C(RPq(2)). Combining methods of topology and operator algebras, we prove that this module is not free. This implies that A cannot be a crossed product of C(RPq(2)) and the integers. To compute the K-theory of A, we present it as a pullback of two copies of the tensor product of the equatorial sphere C*-algebra with the algebra of continuous functions on the interval [0,1]. We put these two copies together by the identity and antipodal automorphisms of the quantum sphere applied at 0 and 1, respectively. Hence A can be viewed as defining a non-trivial bundle over a circle with the fibre being the equatorial quantum sphere. Since replacing the quantum sphere with the unit circle and the antipodal action with the complex conjugation would yield the Klein bottle, we call A the C*-algebra of the Klein-Podles bottle. (Joint work with P.F. Baum.)
Zostanie pokazane twierdzenie mówiące o tym jak wygląda warunkowy drugi moment wolnych rozkładów Meixnera. W tym celu zostanie pokazana rekursja łącząca momenty splotu wolnych rozkładów Meixnera z pewnymi ciągami wyznaczającymi warunkowy drugi moment. Zostanie również przywołane twierdzenie 3.2 Bożejko-Bryc z pracy „On a class of free L´evy laws related to a regression” problem mówiące o tym, że przy założeniu kwadratowej funkcji warunkowej wariancji (plus liniowy pierwszy moment) otrzymujemy wolne rozkłady Meixnera.
W teorii wolnego prawdopodobieństwa Voiculescu i Speichera istnieje "prawo dodawania", które można stosować do hermitowskich macierzy losowych, aby obliczyć rozkład prawdopodobieństwa wartości własnych ich sumy. Opiera się ono zasadniczo na rzeczywistości wartości własnych, a podstawowe role grają w nim funkcje zespolone - Green'a i Blue (inaczej: rezolwenta i R-transformata). Dla niehermitowskich macierzy losowych - fascynujących matematycznie i ważnych w wielu zastosowaniach fizycznych - wartości własne są zespolone i "zespolone" prawo dodawania nie jest w stanie ich opisać. W czasie seminarium pokażę, jak rozwiązać ten problem przez wprowadzenie kwaternionowych funkcji Green'a i Blue oraz "kwaternionowe prawo dodawania". Pokażę też - i zilustruję na przykładach - jak daje ono prostą i czysto algebraiczną metodę rozwiązywania wielu ciekawych sum niehermitowskich macierzy losowych.
We start by considering the classical Jacobi matrices approach to Hamburger moment problems and Pade approximants to Nevanlinna functions (i.e. the Cauchy transform of probability measures). Then we extend the approach to the case of multipoint Pade approximants and Nevanlinna-Pick problems. By using this approach, we get the locally uniform convergence of multipoint Pade approximants to the original function. Finally, we generalize the scheme to the case of Pade approximants for rational perturbations of Nevanlinna functions. This also allows us to prove some convergence results.
I will describe and motivate a natural construction which associates to every finitely generated group G a compact quantum group QISO(G) which can be thought of as the quantum isometry group of the dual of G. QISO(G) has been so far explicitly computed only for some special cases. In this talk I will focus on two examples: a) free groups b) free products of cyclic groups I will explain how the investigation of the new quantum groups turns out to be closely connected with some free probabilistic arguments and categories of noncrossing bi-coloured partitions. This in a sense generalises the work on liberated quantum groups studied recently by Teodor Banica, Roland Speicher and others.
We show that addition of strongly matricially free random variables leads to the `matricial R-transform' related to the associated convolution. It is a linear combination of Voiculescu's R-transforms in free probability with coefficients given by internal units of the considered array of subalgebras. This allows us to view the associated linearization formula as the `matricial linearization property' of the R-transform. Since strong matricial freeness unifies the main types of noncommutative independence, the matricial R-transform plays the role of a unified noncommutative analog of the logarithm of the Fourier transform for the main types of noncommutative independence.
We study two types of C*-algebras that can be constructed canonically from the ring of integers in a number field. The algebras of the first type are simple and have a very interesting structure from the C*-algebra point of view. The second algebra is a Toeplitz type extension of the first one and has better functoriality properties. It also carries a natural one-parameter automorphism group with a KMS-structure which is closely related to number theoretic phenomena.
We show that addition of strongly matricially free random variables leads to the `matricial R-transform' related to the associated convolution. It is a linear combination of Voiculescu's R-transforms in free probability with coefficients given by internal units of the considered array of subalgebras. This allows us to view the associated linearization formula as the `matricial linearization property' of the R-transform. Since strong matricial freeness unifies the main types of noncommutative independence, the matricial R-transform plays the role of a unified noncommutative analog of the logarithm of the Fourier transform for the main types of noncommutative independence.
W referacie rozpatrujemy operatory kreacji, anihilacji i neutralny na zdeformowanej symetrycznej przestrzeni Focka, dla symetryzatora skonstruowanego przy pomocy deformacji odwzorowania (x, y) --> (y, x), danej przez jądro hermitowskie o module 1. Przy ich pomocy konstruujemy nieprzemienne procesy Gaussa i Poissona, a także uogólnione procesy Levy'ego. Pokażemy podstawowe własności tych procesów. Referat oparty jest na wspólnej pracy z M. Bożejko i E. Lytvynovem (Swansea).
Przedstawiona zostanie konstrukcja splotow uogólnionych oparta na klasie miar słabo stabilnych względem splotów uogólnionych, które należą do odpowiedniego obszaru przyciągania. Teoria zostanie poparta ciekawym przykładem splotu Kendalla czy q-splotu w niekomutatywnej probabilistyce.
After recalling the basic facts from the harmonic analysis of the automorphism group of a locally finite, homogeneous tree, we will discuss how important ideas of the harmonic analysis of the symmetric group can be used to study spherical functions and spherical representations of Olshanski spherical pairs consisting of groups of automorphisms of homogeneous trees of infinite order.
In classical probability theory there is known following theorem (proved by Bobecka and Wesolowski): Let U and V be independant real random variables. Assume that $E(V(1-U)|UV)=c$ and $E(V^2(1-U)^2|UV)=d$ for some real constans c,d. Then V has gamma distribution and U has Beta distribution. I will state an analogue of this problem in free probability theory, and give a solution for it.
By looking at a deformed particle density for a quasi-free representation of the Canonical Anticommutation Relations, we will derive a class of determinatal point processes whose correlation kernel is symmetric in an indefinite metric.