# Overview

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## Birkhoff Conjecture for convex planar billiards

### Tuesday, January 22, 2019, 3:00 pm

Centre de recherches mathématiques

Pavillon André-Aisenstadt, Université de Montréal

Room 5345

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.

## Can one hear the shape of a drum and deformational spectral rigidity of planar domains

### Wednesday, January 23, 2019, 3:00 pm

Centre de recherches mathématiques

Pavillon André-Aisenstadt, Université de Montréal

Room 5345

M. Kac popularized the following question "Can one hear the shape of a drum?" Mathematically, consider a bounded planar domain Ω ⊆ **R**^{2} 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. Figalli.

## Stochastic diffusive behavior at Kirkwood gaps

### Friday, January 25, 2019, 4:00 pm

Centre de recherches mathématiques

Pavillon André-Aisenstadt, Université de Montréal

Room 1360

This lecture is aimed at a general audience.

One of the well-known indications of instability in the Solar system is the presence of Kirkwood gaps in the Asteroid belt. The gaps correspond to resonance between their periods and the period of Jupiter. The most famous ones are period ratios 3:1, 5:2, 7:3. In the 1980s, J. Wisdom and, independently, A. Neishtadt discovered one mechanism of creation for the 3:1 Kirkwood gap. We propose another mechanism of instabilities, based on an a priori chaotic underlying dynamical structure. As an indication of chaos at the Kirkwood gaps, we show that the eccentricity of Asteroids behaves like a stochastic diffusion process. Along with the famous KAM theory this shows a mixed behavior at the Kirkwood gaps: regular and stochastic. This is a joint work with M. Guardia, P. Martin and P. Roldan.