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 2013
- Donnerstag, 17.1.2013, 16-18h, SR6 (216) Pierre Tarrago
Noncommutative Symmetric Functions
This talk aims to briefly introduce to a noncommutative generalization of the Hopf algebra of the symmetric
functions. These algebra shows a lot of similarities with the commutative one, going from algebraic to
representation properties. Time permitting I will also present a new kind of generalization containing
the old one.
- Mittwoch, 20.2.2013, 16-18h, SR6 (216) Marek Bozejko
New Generalized Gaussian processes and free infinitely divisibility for classical Meixner distributions
Abstract: In my talk we will present the following subjects:
1. Generalized Gaussian processes related to the functions on pair-partition - the number of singletons
and others.
2.Connections with Markov random matrices as in the paper of Bryc,Dembo,Jiang.
3.Positive definite functions and normson permutation group.
4.Free infinitely divisibility of 1/cosh law and others symmetric Meixner distributions connected with
Meixner-Pollaczek polynomials.
5.Noncommutative Levy processes of anyonic type ; q-Gaussian for |q|=1 and related Hecke deformations.
References:
1.M.Bozejko , W.Bozejko, Generalized Gaussian processes and relations with random matrices and positive
definite functions on permutations groups, arXiv 2013.
2.M.Bozejko, E.Lytvynov, J.Wysoczanski, Noncommutative Levy Processes for Generalized(Particularly Anyon)
Statistics, Comm.Math.Phys. 313(2012) 535-569.
3.M.Bozejko, Deformed Fock spaces, Hecke operators and Monotone Fock space of Muraki, Demonstratio
Mathematica, XLV, 2012, 399-413
- Mittwoch, 27.2.2013, 16-18h, SR6 (216) Sören Möller (Odense, Denmark)
A law of large numbers for the free multiplicative convolution - distributions and observations
In classic probability the multiplicative law of large numbers follows from the additive as a corollary.
This is not the case in free probability, so although an additive law has been proved in [Lindsay-Pata, 1997]
the multiplicative law was only proved recently in [Tucci, 2010] for measures on the positive real line
with compact support. In 2012, in joint work with Uffe Haagerup, we gave a new proof of the result removing
the assumption of compact support. In this talk I will present the result and discuss distributions that
arise in this context and other observations related to the result.
- Dienstag (!), 9.4.2013, 14h ct, SR6 (217) Gaetan Borot, MPIM Bonn
Analytic combinatorics of maps of all topologies
Maps are discrete surfaces obtained by gluing polygons along their edges. The enumeration of rooted planar maps has first been addressed by Tutte in the 60s, thanks to a recursive decomposition of maps which translates into a quadratic functional equation for the generating series of rooted planar maps, which can be solved explicitly. By similar methods, one can derive other (again quadratic) functional equations for the generating series of maps of higher topologies, which are solved by a universal "topological recursion" formula. The initial data of the recursion consists of the generating series of disks (genus 0, 1 rooted boundary) and of cylinders (genus 0, 2 rooted boundaries). Universal here means that the same "topological recursion" also enumerate maps carrying certain statistical physics model (like the Ising model, or the O(n) loop model), provided the initial data is properly chosen. In this talk, I will present the solution of the enumeration problem by the topological recursion for a class of O(n) loop models. This is partly based on joint work with B. Eynard and N. Orantin.
By definition of the model above, it is allowed to identify two edges of the rooted boundary. Then, it is possible to deduce the generating series of maps with simple boundaries, i.e. for which two edges of the same boundaries cannot be identified. At the planar level, the relation between the two generating series is the R-transform. This relation can be explicitly described for all topologies, and it enjoys algebraic properties which still need to be elucidated. In the second part of the talk, I will present these (yet unpublished) observations.
- 10.4.2013, Claus Koestler
The Thompson group F from the viewpoint of noncommutative probability
We characterize the extremal characters of the Thompson group F. Our approach is inspired from techniques in the context of distributional symmetries in noncommutative probability. In particular we address noncommutative independence in the left regular representation of the Thompson group F. This is joint work with Rolf Gohm.
- 29.5.2013, Takahiro Hasebe
Free infinite divisibility for beta distributions
Many beta distribution of first and second kind are shownto be feely infinitely divisible. This generalizes the result of Arizmendi and Belinschi. Taking limits, we can show that many gamma and inverse gamma distributions are freely infnitely divisible too.
- 19.6.2013, Rosaria Simone
Universality results for polynomials in freely independent random variables
The aim of the talk is to provide an overview of the universality results available for polynomials in freely independent random variables, with comparison with the classical probability counterparts. It is known that homogeneous sums based on a sequence of freely independent semicircular elements behave universally with respect to semicircular approximation. Thanks to a general invariance principle and to the celebrated Fourth Moment Theorem, other examples of universal laws are given, with respect to both semicircular and free Poisson approximation, and in both a unidimensional and multidimensional setting. Finally, new persepectives about a possible characterization of universal laws, based on a fourth moment condition, are presented.
- 26.6.2013, Cedric Schonard
Free monotone transport
In 2012 A. Guionnet and D. Shlyakhtenko have shown that there exists a non-commutative analog of Brenier's monotone transport theorem. We can view such a free monotone transportation map as certain kind of replacement of density of a non-commutative law. In this talk I want to present the main ideas of their paper and I will discuss some applications.
- 11.9.2013, 14h ct, Hörsaal III (!), Xiao Xong (Université de Franche-Comté Besançon).
Sobolev inequalities on quantum tori.
We consider quantum tori with a series of operators, called partial derivatives. In analogy to the classical case, we define the corresponding Sobolev spaces and prove the Sobolev embedding inequalities.
- 23.10.2013, Moritz Weber (Universität des Saarlandes, Saarbücken).
The complete classification of easy quantum groups.
Quantum groups generalize the notion of groups in noncommutative operator algebraic settings. The easy quantum groups in turn have a very intrinsic combinatorial component. Its classification (in the orthogonal case) has recently been completed by Sven Raum and myself, and I will report on these results.
- 30.10.2013, Ion Nechita (Université de Toulouse).
Optimization over random subspaces of matrices and free compression norms.
Computing S_1 -> S_p norms of generic quantum channels (completely positive, trace preserving maps) is shown to be equivalent to some optimization problems for norms appearing in free probability. We shall discuss two such examples and describe the relation to free compression norms. If time permits, some applications to quantum information theory will be presented. This is joint work with S. Belinschi, B. Collins and M. Fukuda.
- 06.11.2013, Pierre Tarrago (Universität des Saarlandes, Saarbücken).
Teilklassifizierung der easy Quantengruppen im unitären Fall.
Easy Quantengruppen sind spezielle Quantengruppen, deren Darstellungstheorie kombinatorische Eigenschaften hat. Aufgrund dieser Eigenschaften ist es möglich, eine Klassifizierung dieser Quantengruppen anzugehen. In diesem Vortrag wird eine Klassifizierung im Fall der sogenannten freien Quantengruppen vorgestellt. Dies ist eine gemeinsame Arbeit mit Moritz Weber.
- 04.12.2013, Amaury Freslon (Paris, Frankreich).
Quantum symmetries of noncrossing partitions
The proofs of De Finetti type theorems reveal that the combinatorics of certain joint distributions of (conditionally) free random variables is exactly the same as the combinatorics governing the representation theory of some quantum groups. This leads to the notion of "partition (or easy) quantum group" defined by Banica and Speicher. In this talk, I will take this point of view as far as possible, explaining how a comprehensive (and purely combinatorial) study of noncrossing partitions can yield important results concerning quantum groups. This is based on a joint work with M. Weber.
- 11.12.2013, Mireille Capitaine (CNRS, Institut de Mathematiques de Toulouse).
Exact separation phenomenon for the eigenvalues of large Information-Plus-Noise type matrices. Application to spiked models
- 11.12.2013, Francois Chapon (Paris, Frankreich).
Quantum random walks and matrix Brownian motion.
We will see the construction of non-commutative discrete time approximation of Hermitian Brownian motion by considering quantum random walks, and how this construction allows to understand the Markov property or not of some of its minors eigenvalues processes.
- 11.12.2013, Cedric Schonard (Universität des Saarlandes).
Topics in Free Transportation.
A powerful result by Brenier states that for sufficiently "nice" probability measures the unique solution of Kantorovich's optimal transportation problem for quadratic cost is obtained as the push-forward via the gradient of a convex function. A. Guionnet and D. Shlyakhtenko proved an analogue result in the context of free probability. I will explain the main ideas of their proof and compare the obtained results to Brenier's theorem.
- 11.12.2013, Jonas Wahl (Universität des Saarlandes).
Haagerups Approximationseigenschaft für Quantum Reflection Groups
Assoziiert man zu einer kompakten Quantengruppe ihre reduzierte C* - Algebra, sowie deren einhüllende von Neumann - Algebra, so ergeben sich eine Vielzahl von Fragen nach den Eigenschaften dieser Objekte. Besonders interessant ist dabei die von U. Haagerup eingeführte Approximationseigenschaft (HAP) für finite von Neumann - Algebren, da diese alleine von der Darstellungstheorie der Quantengruppe abhängt. Ich präsentiere einen Beweis der HAP für die Quantengruppe S_n^+ sowie für die Quantum Reflection Groups H_n^s+, welche eng mit den easy Quantengruppen verwandt sind.
Aktualisiert am: 6. November 2017 Moritz Weber