Probabilités, Physique Mathématique et Analyse (2PMA)

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.

TBA

This talk explores the intriguing realm of scattering resonances within two-dimensional transparent cavities, which arose in the modeling of micro-resonators constructed from dielectric materials (with positive permittivity) or metallic nanoparticles (with negative permittivity). Specifically, our investigation is focused on resonances that closely align with the real axis, characterized by highly oscillatory behavior and localization along the interface separating the cavity from its external environment. Notable exemplars of such resonances include whispering-gallery modes observed in dielectric cavities and surface plasmon waves associated with metallic particles.