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:
Kac popularized the following question 'Can one hear the shape of a drum?' Mathematically, consider a bounded planar domain Ω ⊆ R2 with a smooth boundary and the associated Dirichlet problem
Δu + λu=0, u|∂Ω=0.
The set of λ's for which this equation has a solution is called the Laplace spectrum of Ω. Does the Laplace spectrum determine Ω up to isometry? In general, the answer is negative. Consider the billiard problem inside Ω. Call the length spectrum the closure of the set of perimeters of all periodic orbits of the billiard inside Ω. Due to deep properties of the wave trace function, generically, the Laplace spectrum determines the length spectrum. We show that a generic axially symmetric domain is dynamically spectrally rigid, i.e. cannot be deformed without changing the length spectrum. This partially answers a question of P. Sarnak. The talk is based on two separate joint works with J. De Simoi, Q. Wei and with J. De Simoi, A. Figal.
