Research seminar Free Probability Theory

In this research seminar we treat topics ranging from free probability and random matrix theory to combinatorics, operator algebras, functional analysis and quantum groups.

See here for recent talks

Talks in 2012

• Dienstag (!), 10.01.2012, 16 Uhr ct, HS IV       Claus Köstler
  Ein operatoralgebraischer Zugang zur Darstellungstheorie der unendlichen symmetrischen Gruppe
  Die unendliche symmetrische Gruppe ist ein typisches Beispiel fuer eine "grosse Gruppe", die keine
  (nicht-trivialen) irreduziblen Darstellungen besitzt. Thoma bestimmte erstmals in den 60er Jahren ihre
  extremalen Charaktere. Alternative Beweise dieses Satzes von Thoma wurden von Vershik & Kerov
  in den 80er Jahren, sowie von Okounkov in den 90er Jahren gegeben. Ich werde in meinem Vortrag
  einen neuen operatoralgebraischen Beweis des Satzes von Thoma vorstellen.
  
  
• 01.02.2012       Natasha Blitvic
  The (q,t)-Gaussian Process
  The setting for this talk is the deformed commutation relations and Fock space representations of deformed
  quantum harmonic oscillator algebras. Building on the work of Bozejko and Speicher in the single-parameter
  case, we will begin by constructing the (q,t)-Fock space, a two-parameter deformation of the Bosonic and
  Fermionic Fock spaces. We will discuss the probablistic interpretation of the algebras of bounded linear
  operators on this space; that is, the ensuing two-parameter continuum of non-commutative probability theories,
  with particular emphasis on the operators playing the role of the classical Gaussian random variables.
  We will focus on the remarkable combinatorial structure underlying these objects and, if time permits, their
  intimate ties to other mathematical entities.
  
  
  
• 08.02.2012       Tobias Mai
  Berry-Esseen estimates for a multivariate free central limit theorem
  We address the question of a Berry-Esseen type theorem for the speed of convergence in a multivariate free
  central limit theorem. For this, we estimate the difference between the operator-valued Cauchy transforms of the
  normalized partial sums in an operator-valued free central limit theorem and the Cauchy transform of the limiting
  operator-valued semicircular element. Since we have to deal with non-self-adjoint operators in general, we
  introduce the notion of matrix-valued resolvent sets and we study the behavior of Cauchy transforms on them.
  
  
• 17.02.2012       Octavio Arizmendi
  Free and Boolean Strictly Stable Laws revisited.
  Infinitely divisible measures play an important role in probability theory, since they arise from very general
  limit theorems. An important subclass of them is the one of stable laws.
  Bercovici and Pata showed that classical, Free and Boolean strictly stable laws are in bijection. In this talk
  we consider different aspects of Free and Boolean Stable laws.
  We start by proving that the Cauchy distribution is the only fixed point of the Boolean-to-Free Bercovici Pata
  Bijection. Next, we show a reproducing property for all free and Boolean stable laws. This generalizes previous
  results by Biane, for positive free stable laws and by Arizmendi and Perez Abreu, for symmetric ones.
  Moreover, this reproducing property implies the free infinite divisibility of Boolean Stable laws, with index
  1/n, for n integer greater than 2.
  Belinschi and Nica introduced a composition semigroup on the set of probability measures B_t. This semigroup
  coincides with the Boolean-to-Free Bercovici Pata Bijection, for t=1.Using this semigroup, they introduced a
  free divisibility indicator, from which one can know whether a probability measure is freely infinitely divisible
  or not.
  Finally, we address the problem of calculating the free divisibility indicator of Boolean and free stable laws.
  In particular we find new examples of probability measures with free divisibility indicator infinity.
  This is a joint work with Takahiro Hasebe.
  
  
• 29.02.2012       Uwe Franz
  What are Brownian motions on compact quantum groups.
  We recall the definition of Levy processes on quantum groups and discuss the problem of characterising Brownian
  motions. We suggest generalisations to compact quantum groups of several desirable properties:
  gaussianity, symmetry, and invariance under the adjoint action.
  While some compact quantum groups, e.g., the Woronowicz quantum group SU_q(2) with q^2\not=1 and the other
  q-deformations of compact simple Lie groups, have no quantum stochastic processes satisfying all these properties,
  it turns out that the liberated orthogonal quantum group O_n^+ has a unique Brownian motion.
  
  
  
• 29.02.2012       Sören Möller
  An overview of (complete) approximation properties for C*-algebras
  There are a number of approximation properties a C*-algebra might have. This talk will give an overview over some
  of them, focusing on implications between properties as well as the special case of group algebras, in which many
  of these properties can be translated to thegroups.
  
  
• 14.03.2012       James Mingo
  The Moebius Function and the Weingarten Function
  Two of the most important functions in free probability are the Moebius function and the Weingarten function.
  The Moebius function is used to write cumulants in terms of moments and the Weingarten function is used
  to calculate integrals against the Haar measure of the unitary group U(n).
  I will explain the background to these functions and show how they are related by a simple equation.
  This is joint work with Roland Speicher.
  
  
• 14.03.2012      Emily Redelmeier
  A Graphical Calculus for Haar-Distributed Orthogonal Matrices and Second-Order Real Freeness
  We present a graphical calculus for computing the expected values of traces of expressions involving
  Haar-distributed orthogonal matrices, similar to the genus expansion for various Gaussian matrices.
  As with the real genus expansions, the expansion includes nonorientable surfaces, and the order of
  a term depends only on its Euler characteristic.
  We use this to demonstrate that independent matrices which are orthogonally in general position are
  second-order real free.
  As a corollary, any orthogonally invariant distribution, including the Haar-distributed matrices themselves,
  are second-order real free.
  
  
• 23.03.2012       Octavio Arizmendi
  Second order even and R-diagonal operators.
  We propose definitions of even and R-diagonal elements of second order, including the semicircular and haar unitaries considered by Mingo and Speicher in connection of fluctuation of Gaussian Unitary Ensembles and Haar distributed random unitaries, respectively. We give a formula for the (2nd order) free cumulants of the square x^2 of a second order even element in terms of the (2nd order) free cumulants of x. We get similar results the free cumulants of aa* when a is second order R-diagonal. We also show that if a is R-diagonal and b is second order free from a then ab is also R-diagonal, showing that second order R-diagonal operators exist in abundance.
  This is a joint work with James Mingo.
  
  
• 24.04.2012       Piotr Sniady
  Trajectiories of jeu-de-taquin and semicircular law
  In an infinite Young tableau we remove the corner box. In this way an avalanche of sliding boxes is
  created. Asymptotic behavior of this avalanche is closely related to Robinson-Schensted-Knuth (RSK)
  correspondence and the representation theory of the infinite symmetric group. Wigner's semicircular law
  plays an important role.
  This is a joint work with Dam Romik.
  
  
• 02.05.2012       Tobias Mai and Soren Moller
  On Wigner chaos and the fourth moment condition
  
  
• 06.06.2012       Octavio Arizmendi Echegaray
  Convergence of the fourth moment for infinitely divisible distributions
  In a seminal paper, Nualart and Peccati (2005),proved that for multiple Wiener integrals with variance 1
  in a fixed chaos, the convergence of the 4th moment to 3 implies the convergence in law to the Gaussian
  distribution. Later, Kemp, Nourdin, Peccati and Speicher (2011) proved the analogous theorem in
  free probability. In this talk we will explain a similar result for both classically and freely infinitely divisible
  measures. Namely, when restricted to classically (resp freely) infinitely divisible distribution, the
  convergence of the first four moments to the Gaussian (resp. Semicircle distribution) implies the
  convergence in law.
  
  
• 13.06.2012       Michael Anshelevich
  Generators of some non-commutative stochastic processes: proofs and further results
  The following results are well known. A Levy process is a Markov process. Its transition operators form
  a semigroup of contractions. The generator of this semigroup can be written down explicitly using
  Fourier analysis.
  I will discuss the same questions in the framework of free Levy processes (process with freely independent,
  stationary increments). Some of the new difficulties in this case are (1) there is no free Fourier
  transform, and (2) the transition operators no longer form a semigroup. Nevertheless, most of the results
  from the first paragraph carry over, with the generators written down explicitly as singular integral
  operators.
  
  
• 12.07.2012       John Williams
  The inclusion of monotone convolution in the Bercovici-Pata bijection
  I will discuss recent work in which the bijection of infinitely divisible measures and triangular arrays
  in the various probability theories is extended to monotone convolution. The key to proving this theorem
  is to treat F-transforms as composition operators and study these phenomenon using tools from semi-group
  theory. In particular, we utilize the famous product formula due to Chernoff. We also prove a limited
  converse to Chernoff's theorem for composition operators on certain spaces of analytic functions.
  
  
• 23.10.2012       Octavio Arizmendi
  Superconvergence in Non-Commutative Probability: A combinatorial approach.
  In this the talk I will explain how to use cumulants to give a simple proof of an instance of the so-called
  superconvergence of normalized sums of free random variables. Namely, that the operator norm of normalized
  sums of bounded free random variables with mean 0 and variance1, converge to 2. Moreover, our approach
  generalizes in a straightforward way to monotone and boolean independence and q-convoluion.
  
  
  
• 12.11.2012       Roland Friedrich
  On certain Algebraic Structures in Free Probability Theory
  In this talk we shall discuss some algebraic structures which are inherently related to free probability
  theory, as we found. This permits us to give a web of correspondences between different but isomorphic
  rings. In particular it extends the known boxed operations in free harmonic analysis, such that the set of
  distributions becomes a commutative unital ring. This will be illustrated by examining some of the most
  important distributions.
  

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Aktualisiert am: 6. November 2017  Moritz Weber

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Fachrichtung Mathematik Universität des Saarlandes