CEMPI Distinguished Lecture Series

Anatole Katok (Penn State University)
Entropy in dynamical systems: complexity, flexibility and rigidity.
Friday September 15, 2017.
Salle de Réunion, M2.

Several interrelated concepts of entropy as well as closely related notions of Lyapunov characteristic exponents play a central role in the modern theory of dynamical systems. Those notions give quantitative expression of the measure of exponential complexity present in a deterministic system. After a general review of those concepts and principal relations between them I will discuss results and open problems related to two complimentary phenomena. of flexibility and rigidity. The general paradigm of flexibility can be rather vaguely formulated as follows: Under properly understood general restrictions within a fixed class of smooth dynamical systems quantitative dynamical invariants take arbitrary values. Precise calculation are possible only in very few cases, primarily of algebraic nature such as homogeneous or affine systems. Most known constructions are perturbative and hence at best would allow to cover a small neighborhood of the values allowed by the model, or more often, not even that, since those models are often ``extremal’’. So establishing flexibility calls for non-perturbative or large perturbation constructions in large families to cover possible values of invariants. On the other hand, there is the rigidity paradigm that is better developed. It has several aspects and in the case of classical systems with discrete and continuous time one of them is related to these quantitative characteristics of exponential complexity: Particular values of entropies or Lyapunov exponents or relations between those determine algebraic or similar models within a broad class of systems. Rigidity becomes more common and even prevalent when one passes from classical systems to systems with multi-dimensional time.

Gilles LEBEAU (Laboratoire J.A. Dieudonné, Nice)
Du Son et de la Lumière: Modélisation et Analyse Mathématique.
Monday, June 26, 2017, 20h.
LILLIAD.

Nous présenterons un rapide survol historique des travaux sur la propagation des ondes depuis Descartes, Newton, Huygens, Euler, Fresnel, Maxwell... Nous rappellerons ensuite l’influence de l’étude des ondes sur l’analyse mathématique: analyse de Fourier, espaces fonctionnels, analyse dans l’espace de phase et géométrie symplectique. Enfin, nous terminerons par des problèmes ouvert sur les ondes non linéaires, et quelques applications: Problèmes inverses, Contrôle non destructif, Photonique.

Herbert Spohn (Technical University Münich)
Nonlinear fluctuating hydrodynamics and time-correlations for one-dimensional systems.
Friday June 9, 2017, 11h15.
salle des Séminaires, M2.

Our focus are non-integrable classical systems in one dimension, like Fermi-Pasta-Ulam chains, the discrete nonlinear Schroedinger equation, and 1D fluids interacting through a short range potential. Nonlinear fluctuating dynamics is used to compute and predict their time-correlations in thermal equilibrium.We will explain the connection to the multi-component Kardar-Parisi-Zhang equation.

Carlangelo Liverani (Université Roma 2 Tor Vergata)
Isolated systems and noise: a persistent (and useful) illusion.
Friday January 27, 2017, 11h.
salle des Séminaires, M2.

I will discuss, via the rigorous analysis of a seemingly super simple example, how the appearance of isolated systems and noise emerges in classical fast-slow dynamical system.

Henri Darmon (Université McGill et Paris 7)
La conjecture de Birch et Swinnerton-Dyer: progrès et questions ouvertes.
Thursday December 10, 2015, 11h.
salle Kampé de Fériet, M2.

Cet exposé non technique parlera de certains progrès récents (dus entre autres à Bhargava, Skinner, Venerucci, et Wei Zhang) et moins récents (travaux de Gross-Zagier-Kolyvagin et de Kato) sur la conjecture de Birch et Swinnerton Dyer, pour faire le bilan de ce que l'on connait (et aussi, de ce que l'on ne connait pas) sur cette conjecture célèbre qui figure parmi les sept ''problèmes du millénaire'' proposés par l'institut Clay.

T. Ransford (Université de Laval)
Spectres et pseudospectres.
Friday October 17, 2014, 11h15.
salle de Réunion, M2.

Les valeurs propres sont parmi les notions les plus utiles en mathématiques: elles permettent la diagonalisation des matrices, elles décrivent l'asymptotique et la stabilité, elles donnent de la personnalité à une matrice. Cependant, lorsque la matrice en question n'est pas normale, l'analyse par des valeurs propres ne donne qu'une information très partielle, et peut même nous induire en erreur. Cet exposé se veut une introduction à la théorie des pseudospectres, un raffinement de la théorie spectrale standard qui s'est avéré utile dans des applications concernant des matrices non normales. Je vais m'intéresser surtout à la question suivante: A quel point les pseudospectres d'une matrice déterminent-ils le comportement de la matrice?