Mathematical INstitute @ Tel Aviv

Coordinators

• Noga Alon
• Michael Krivelevich
• Leonid Polterovich

The Institute supports a variety of research-related activities at the School of Mathematical Sciences of TAU, including distinguished lectures, visitors, organization of workshops and travel of research students.

• Yakov Eliashberg (Stanford University)
• Peter Sarnak (Princeton University and IAS)
• Alain-Sol Sznitman (ETH, Zürich)

Vadim Kaloshin (University of Maryland)

Date Day Time Location Title
2017-06-05 Mon 12:15 Schreiber 006 Can you hear the shape of a drum and deformational spectral rigidity of planar domains
2017-06-07 Wed 14:10 Schreiber 309 Birkhoff Conjecture for convex planar billiards

Vadim Kaloshin (University of Maryland)

Date Day Time Location Title
2017-06-05 Mon 12:15 Schreiber 006 Can you hear the shape of a drum and deformational spectral rigidity of planar domains
2017-06-07 Wed 14:10 Schreiber 309 Birkhoff Conjecture for convex planar billiards

Abstracts:

Can you hear the shape of a drum and deformational spectral rigidity of planar domain

M. Kac popularized the question `Can you hear the shape of a drum?’. Mathematically, consider a bounded planar domain $\Omega$ and the associated Dirichlet problem $\Delta u+\lambda^2 u=0, u|_{\partial \Omega}=0.$ The set of $\lambda$’s such that this equation has a solution, is called the Laplace spectrum of $\Omega$. Does Laplace spectrum determines $\Omega$? In general, the answer is negative.

Consider the billiard problem inside $\Omega$. Call the length spectrum the closure of the set of perimeters of all periodic orbits of the billiard. Due to deep properties of the wave trace function, generically, the Laplace spectrum determines the length spectrum. We show that any axis symmetric planar domain with sufficiently smooth boundary close to the disk is dynamically spectrally rigid, i.e. can’t be deformed without changing the length spectrum. This partially answers a question of P. Sarnak.

This is a joint work with J. De Simoi and Q. Wei.

Birkhoff Conjecture for convex planar billiards

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 Avila, De Simoi, G.Huang, Sorrentino.