Séminaires

Malte Gerhold (Université de la Sarre) : Classification of two-faced independences Abstract: Two-faced independences are independence relations for pairs of noncommutative random variables. An important example is bifree independence, which models the relation between left and right regular representation of the free group in the canonical tracial state. Around 2000, in works of Speicher, Ben Ghorbal & Schürmann, and Muraki, a complete classification of "single-faced" independences was obtained: the only independences in this case are boolean, tensor, free, monotone and anti-monotone independence. I report on the current status of the classification program for two (or multi-faced) independences and we will focus on the interplay between the lifting approach (discussed in the preceding talk, given by Takahiro Hasebe) and the combinatorial "cumulant" approach to multi-faced independences. Based on joint work with Takahiro Hasebe, Michaël Ulrich, and Philipp Varšo.

Takahiro Hasebe (Hokkaido University) : Lifting operators on a Hilbert space nicely to the tensor product or free product Hilbert space Abstract: The basic five notions of independences in noncommutative probability (tensor, free, boolean, monotone, antimonotone) have canonical operator models on Hilbert spaces. All these models are based on constructing a "good composite Hilbert space" (mostly, the tensor product or free product Hilbert spaces) and a "good lifting" of operators to the composite space. In classical probability, this corresponds to the well known construction of independent random variables on the product of probability spaces. In this talk, we axiomatize "good liftings" and classify them. The subsequent talk, given by Malte Gerhold, discusses applications to finding new noncommutative independences for random vectors. This talk is based on a joint work with Malte Gerhold and Michael Ulrich.

Oleg Szehr (Swiss AI Lab IDSIA): New insights on the asymptotic behavior of Jacobi polynomials and applications Abstract: Although a recurrent tool in applied asymptotic analysis, the asymptotic behavior of Jacobi polynomials with varying parameters is frequently good for a surprise. We introduce a new, simple and robust approach to this topic and demonstrate that certain "established" formulas in the literature yield inaccurate results. Using our method we identify previously unknown forms of asymptotic behavior and we investigate how this affects applications of Jacobi polynomial asymptotics: 1) We show how the Fourier coefficients of powers of Blaschke factors can be written in terms of Jacobi polynomials and we discuss their asymptotic behavior. 2) We discuss the asymptotics of quantum random walks on 1D spin systems using Jacobi polynomials, where we observe a new form of asymptotic behavior. This talk is based on: https://www.sciencedirect.com/science/article/pii/S0021904522000041

Hao Zhang : Schatten class membership of commutators Abstract: Commutators are well-known as generalization of Hankel type operators. In this talk, we are concerned about the Schatten class membership of commutators involving nondegenerate singular integral operators. We describe their Schatten class membership in terms of Besov spaces. Our approach is based on the complex median method, which is an extension of Lerner’s median method and is applicable to complex-valued functions.

Alexander Volberg : A_2 conjecture fails for matrix weights because 3/2 >1. Abstract: How to estimate the norm of the Hilbert transform (or other singular operators) in space [katex]L^2(w)[/katex] via weight [katex]w[/katex] This classical question appeared from probability (i.e. from the regularity theory of stationary stochastic processes). Initial answer was given by works of Hunt—Muckenhoupt—Wheeden and Helson—Szeg\”o in the 1970’s. But the sharp estimates are from 2000’s due to Petermichl, Volberg, Hytönen. The answer (obtained first for Ahlfors—Beurling transform, then for the Hilbert transform, and then for all Calder\’on—Zygmund operators) is that the bound is linear in [katex] A_2[/katex] Hunt—Muckenhoupt—Wheeden characteristic of weight [katex]w[/katex] . This was a solution of the so-called [katex] A_2[/katex] conjecture. But what if the stationary process in question is a vector one (as Wiener and Masani asked)? Then the weight [katex] W[/katex] is a matrix weight. It was a long standing problem to prove that the same linear estimate holds. However, recently we, Domelevo—Petermichl—Treil—Volberg, showed that this is not the case, [katex]A_2[/katex] conjecture fails, and the right sharp estimate is [katex] [W]_{A_2}^{3/2}[/katex] .

Hering, Müller-Lyer, Ebbinghaus, Wundt & Fick, Ponzo, Oppel & Kundt, Kanizsa, Jastrow… sont tous des scientifiques qui ont associé leur nom à de célèbres illusions d’optique.
L’exposé, principalement basé sur des illustrations, montre comment le cerveau peut être trompé par des images et tente d’en expliquer le mécanisme.

Lien pour suivre le séminaire à distance : https://webconf.lal.cloud.math.cnrs.fr/b/ste-23y-t9y

Amaury Freslon (Laboratoire de Mathématiques d’Orsay) : A la recherche du mouvement brownien quantique Résumé : La théorie des processus aléatoires sur les groupes quantiques compacts est établie depuis maintenant plusieurs décennies, mais notre compréhension des exemples concrets reste très partielle. Par exemple, dans le cas des groupes quantiques de matrices - qui généralisent les groupes de Lie - il n'y a pas à ce jour de notion générale de mouvement Brownien. Je présenterai une première approche à ce problème basée sur la notion de processus Gaussien due à Schürmann, qui mène à des questions topologiques intrigantes. J'en présenterai ensuite une seconde, due à Cipriani, Franz et Kula, qui mène à des exemples très concrets dont ont peut explorer les propriétés asymptotiques.

In this talk we present some results about the existence of normalized ground states on noncompact metric graphs for nonlinear Schrödinger equations involving possibly both a standard power nonlinearity and delta nonlinearities located at the vertices of the graph. In the first part, we review some results when the sole standard nonlinearity is present. In the second part, we present more recent results, both when only delta nonlinearities at the vertices are considered and when standard and delta nonlinearities cohexist. In the first case, we show that the ground state problem strongly depends on the degree of periodicity of the graph, the total number of delta nonlinearities and their dislocation in the graph. In the second case, we highlight that the existence of ground states is strongly affected by the value of the mass, the relation between the powers of the two nonlinear terms and by topological and metric properties of the graph itself. These results have been obtained in collaboration with R. Adami, S. Dovetta and E. Serra (Politecnico di Torino).

Je discute quelques exemples de démonstrations mathématiques et l'on se pose des questions quant à leur force de conviction (telle démonstration est-elle vraiment convaincante ?), et quant à la beauté ou l'utilité du résultat.
Il n'est nul besoin pour cela de faire appel à des mathématiques considérées comme difficiles. Une fois la discussion lancée, les participants pourront proposer les exemples qu'elles considèrent comme plus ou moins troublants.
La thèse défendue (en suivant Bishop) est que les mathématiques sont avant tout une question de «sens commun» et que seule la discussion contradictoire au sein de la communauté mathématique permet de «mettre les choses au clair», autant que faire se peut.

Le lien pour suivre le séminaire à distance reste le même : https://webconf.lal.cloud.math.cnrs.fr/b/ste-23y-t9y

Tao Mai (Baylor University, Texas) : An extension of Hilbert transform to hyperbolic groups Abstract: The classical Hilbert transform has a natural analogue on the nonabelian free groups by decomposing the free group into disjoint subsets according to the first letter of the reduced words. Mei and Ricard prove that such a transform (decomposition) is unconditional with respect to the noncommutative Lp norm associated with the free group von Neumann algebras for all 1<p<\infty. I plan to talk about a possible extension of the Lp unconditionality of such “transforms” to the general hyperbolic groups.