- 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.
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