Conférence Nirenberg en analyse géométrique
Conférence de Vadim Kaloshin, titulaire de la chaire Michael Brin en mathématiques de l'Université du Maryland. Il a obtenu son doctorat de l'Université de Princeton en 2001 sous la supervision de John Mather. Le professeur Kaloshin a apporté des contributions fondamentales à la théorie des systèmes dynamiques, notamment à l'étude de la diffusion d'Arnold, du problème à n corps et de la conjecture de Birkhoff pour le billard convexe.
Résumé:
G.D. Birkhoff introduced a mathematical billiard inside of a convex domain as the motion of a massless particle with elastic reflection at the boundary. A theorem of Poncelet says that the billiard inside an ellipse is integrable, in the sense that the neighborhood of the boundary is foliated by smooth closed curves and each billiard orbit near the boundary is tangent to one and only one such curve (in this particular case, a confocal ellipse). A famous conjecture by Birkhoff claims that ellipses are the only domains with this property. We show a local version of this conjecture —namely, that a small perturbation of an ellipse has this property only if it is itself an ellipse. This is based on several papers with A. Avila, J. De Simoi, G. Huang and A. Sorrentino.
