The Birkhoff Conjecture, Spectral Rigidity for Convex Reflecting Particle Systems, and Stochastic Arnold Diffusion

University Of Maryland, College Park, College Park MD

Investigators

Abstract

Analysis of the motion of a particle moving inside of a convex domain without friction and reflecting elastically against the walls was initiated by mathematician G.D. Birkhoff in 1927. This mathematical dynamical system serves as a good approximation to several important physical systems, for instance, sound propagation inside of a chapel or a cathedral. This research project targets classical inverse problems for such dynamical systems, for example: if we know some properties of the particle's motion, can we infer properties of the underlying domain? This is closely related to the question, "Can you hear the shape of a drum?" That is, can one reconstruct the shape of a drum solely from the sound it produces? It turns out that properties of sound inside of a convex domain are closely related to properties of the possible trajectories of the associated reflecting particles inside of the same domain. This project aims to deepen understanding of related mathematical questions in the analysis of dynamical systems. The first part of the project is to study perturbations of convex billiards and to derive necessary conditions for integrability (for perturbation of ellipses) and for isospectrality of deformations of convex domains. The former condition is closely related to the phenomenon of whispering galleries and the Birkhoff conjecture about characterization of integrable billiards, while the latter problem is closely related to deformation spectral rigidity. M. Kac's question can be expressed in terms of the spectrum of the Dirichlet problem for the Laplace equation inside of a convex planar domain. Due to the wave trace function the Laplace spectrum generically determines the length spectrum of the associated billiard inside of the same domain and connects analysis with billiards. The second part of the project is about proving stochastic Arnold diffusion, that is, stochastic diffusing behavior for nearly integrable systems and the Chirikov conjecture. One of challenging goals is to establish stochastic diffusive behavior inside of Kirkwood gaps for the three-body problem.