Prof. Dr. Roland Speicher

Prof. Dr. Moritz Weber

Miguel Pluma

Research seminar Free Probability Theory

Oberseminar zur Freien Wahrscheinlichkeitstheorie

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

Time and place

Wednesdays, 16:15-18:00, room SR 6, building E2 4
talks are 60 minutes plus discussion

Talks in 2019

  • 23.10.2019,  16:00 c.t.,   Christian Voigt    (Glasgow, Scotland)
    The Plancherel formula for complex quantum groups
    The Plancherel formula plays a basic role in Fourier analysis and its applications, and harmonic analysis in general. From an operator algebra point of view it is a useful tool to understand the structure of certain (group) C^*-algebras. Starting from the classical Fourier transform, I'll review several versions of the Plancherel Theorem, in increasing levels of generality. Then I'll discuss a specific example, namely the Plancherel formula for a certain class of deformations of complex Lie groups like SL(n, \mathbb{C}), including some background and motivation.

  • 17.07.2019,  16:00 c.t.,   Chase Vogeli    (MIT, USA)
    Some applications of graph theory to the study of quantum symmetry
    In this talk, I will describe two situations in which graph theory can be applied to understanding quantum symmetry. First, I will discuss uniformly vertex-transitive graphs and some methods for their detection. Furthermore, we demonstrate that there exist vertex-transitive graphs which are not uniformly vertex-transitive, settling a previously open question. This was motivated by recent work by Nechita, Schmidt, and Weber, who developed an algorithm which can find quantum symmetries in uniformly vertex-transitive graphs. Secondly, I will describe a work-in-progress project with Daniel Gromada which concerns how understanding the representations of a quantum analog of the Coxeter group D_4 can be reduced to enumerative questions about finite graphs.

  • 10.07.2019,  16:00 c.t.,   Michael Fleermann    (FernUni Hagen)
    Global and Local Semicircle Laws for Random Matrices with Correlated Entries
    We analyze ensembles of random matrices with correlated entries, for which we derive global and local semicircle laws. Global semicircle laws can estimate the fraction of random eigenvalues on fixed intervals. In contrast, local semicircle laws may conduct these estimations dynamically on a sequence of intervals which exhibit an appropriate diameter decay. The global laws are derived in probability and almost surely for full and band random matrices with correlations, admitting correlated Gaussian and Curie-Weiss entries as examples. The local laws are derived for full matrices with Curie-Weiss entries.

  • 10.07.2019,  15:00 s.t.,  Mario Diaz    (CIMAT, Mexico)
    Analysis of neural networks using free probability techniques
    Among all the types of neural networks available nowadays, deep linear neural networks (DLNNs) are perhaps the easiest to implement. Nonetheless, there is empirical evidence showing that their learning dynamics are similar to those of their non-linear counterparts. In this talk we present a basic learning setup where DLNNs are applied and review an analysis of their learning dynamics based on random matrix techniques ,as done by Liao and Couillet. We finish presenting a new take to this problem using free probabilistic ideas. This is work in progress with Carlos Madrid (Universidad de Guanajuato) and Víctor Pérez-Abreu (CIMAT).

  • 22.05.2019,  Xumin Wang    (Besacon, France)
    Examples of spectral triples: spheres
    I will talk about the spectral triples for classical, half-liberated and free spheres. By determining eigenvalues and eigenvectors spaces, we classify the Dirac or Laplacian operators on spheres. According to this spectral triples, the spectral dimension of these spheres can be computed.

  • 13.05.2019,  Johannes Flake    (Aachen, Germany)
    Deligne's interpolation category Rep(S_t) and its monoidal center
    The representations of the symmetric group on n letters form a semisimple tensor category for each natural number n. Pierre Deligne defined a family of categories parametrized by the complex numbers which interpolate those categories in a certain precise sense. I will give an introduction to Deligne's interpolation categories and discuss joint work with Robert Laugwitz about the monoidal center of them yielding, in particular, interpolation objects for Yetter--Drinfeld modules of the symmetric groups.

  • 08.05.2019,  Isabelle Baraquin   (Besacon, France)
    Analysis and probability on the Sekine family of finite quantum group
    In this talk, we will first present a result from Diaconis, Shahshahani and Evans. Let M be a random matrix chosen from the unitary group U(n) and distributed according to the Haar measure. Then, for j∈N, Tr(M^j) are independent and distributed as some complex normal variables when n→∞. We will then look at this type of result in the framework of the Sekine finite quantum groups KP_n. In a second part, we will also study convergence of random walks, on Sekine finite quantum groups, arising from linear combination of irreducible characters. Thanks to Quantum Diaconis-Shahshahani Theory we bound the distance to the Haar state and determine the asymptotic behavior, i.e. the limit state if it exists.

Past talks


Aktualisiert am: 13. May 2019   Miguel Pluma Impressum