Responsable : Martin Averseng
Le séminaire a lieu usuellement le jeudi de 11h à 12h.
Séminaires à venir
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Séminaires passés
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The analysis on R^d equipped with a root system, often called "Dunkl analysis", is motivated by mathematical physics. Dunkl operators occur naturally in the study of quantum particle systems of Calogero-Moser type with interactions on a line or a torus. This analysis includes Dyson Brownian motion, where Brownian particles move under the condition not to collide. The harmonic analysis in the Dunkl and Heckman-Opdam setting is far advanced. However, several classical results on R^d are still open in this setting. After a short general introduction to root systems and related Dunkl analysis, we will present our recent results with K. Stempak: Let R be a root system in \mathbb{R}^d. Let $C_+\subset \mathbb{R}^d$ denote a positive Weyl chamber distinguished by a choice of R_+, a set of positive roots. We define and investigate Hardy and BMO spaces on C_+, with boundary conditions given by a homomorphism which attaches the +/- signs to the facets of C_+. Specialized to orthogonal root systems, atomic decompositions in Hardy spaces are obtained and the duality problem is also treated.
We indicate the structure of automaton with multiplicities (taken in a non necessarily commutative semiring R), following the original thought of Eilenberg and Schützenberger. Computation of its behaviour, a generating series, entails that of the star of a matrix (in general with noncommutative coefficients). When one specializes R to R = Sigma x K (K a ring of operators), one gets a powerful notion of Sigma-action which is powerful enough to, for example, generate Hyperlogarithms and, through Lazard elimination, explain the asymptotic initial conditions, and through Radford's theorem, transcendance results. When one specializes R to R= Sigma x K (K a commutative semiring), one gets the classical structure of automaton with multiplicities in K, and if, moreover, K is a field, one can use this rational calculus to compute within every Sweedler's dual of a K Hopf or bi-algebra. If time permits we will describe an unexpectedly simple two-state ‘‘letter-to-letter'' transducer which produces the Collatz function which opens the door to a geometrization of the Collatz conjecture.
L'objectif de cet exposé est de décrire, dans la limite semi-classique, la propagation de fonctions d'onde le long d'une interface entre deux isolants topologiques en deux dimensions. Nous supposerons que cette interface est une courbe lisse connexe sans bords. Nous considérerons un système d'équations d'évolution régi par une modulation adiabatique d'un opérateur de Dirac (non magnétique) de masse variable s'annulant à l'interface. Nous décrirons la propagation des solutions de ce système en termes de mesures semi-classiques, en utilisant une procédure de forme normale et une seconde microlocalisation proche de l'interface.
The objective of this work is to study how wave propagates in the frequency regime in quasi-periodic media. Quasi-periodic media are characterized by quasi-periodic coefficients, i.e. functions that are the value in a certain (irrational) direction of periodic functions in a higher dimension. Quasi-periodic media are somehow between periodic media and random media: they have a structure without being periodic and they exhibit properties well known in random media such as ergodicity or localization. I want to explain in this talk how we can take advantage of their structure to define, when it is possible, the outgoing/physical wave and a notion of group velocity.
