In 1966, Marc Kac in his famous paper 'Can one hear the shape of a drum?' raised the following question: Is a bounded Euclidean domain determined up to isometries from the eigenvalues of the Euclidean Laplacian with either Dirichlet or Neumann boundary conditions? Physically, one motivation for this problem is identifying distant physical objects, such as stars or atoms, from the light or sound they emit.

The only domains which are known to be spectrally distinguishable from all other domains are balls. It is not even known whether or not ellipses are spectrally rigid, i.e. whether or not any continuous family of domains containing an ellipse and having the same spectrum as that ellipse is necessarily trivial.

In a joint work with Steve Zelditch we show that ellipses are infinitesimally spectrally rigid among smooth domains with the symmetries of the ellipse. Spectral rigidity of the ellipse has been expected for a long time and is a kind of model problem in inverse spectral theory. Ellipses are special since their billiard flows and maps are completely integrable. It was conjectured by G. D. Birkhoff that the ellipse is the only convex smooth plane domain with completely integrable billiards. Our results are somewhat analogous to the spectral rigidity of flat tori or the sphere in the Riemannian setting. The main step in the proof is the Hadamard variational formula for the wave trace. It is of independent interest and it might have applications to spectral rigidity beyond the setting of ellipses. The main advance over prior results is that the domains are allowed to be smooth rather than real analytic. Our proof also uses many techniques developed by Duistermaat-Guillemin and Guillemin-Melrose in closely related problems.

We consider the spreading of a droplet on a planar surface covered
with random obstacles. Assuming the obstacles are stationary ergodic
and taller than the droplet, we show that the homogenized limit is
described by a droplet spreading on a flat surface but with a reduced
front speed and surface tension. This is joint work with Inwon Kim.

The continuity of Lyapunov exponent plays an important role for many problems in quasi-periodic cocycles. One example is Ten Martini problem. It is well known that the Lyapunov exponent is continuous in analytic topology and discontinuous in C^0-topology. In this talk, we will provide quasi-periodic cocycles at which the Lyapunov exponent is not continuous in C^l-topology with 0 ≤ l ≤ +∞. This is joint work with Jiangong You.

The parity conjecture is a weak version of Birch-Swinnerton-Dyer
Conjecture or more generally, Beilinson-Bloch-Kato Conjecture. It is
conjectured that the vanishing order of the L-function at the central
point has the same parity as the dimension of the Bloch-Kato Selmer
group. I will explain an approach to this conjecture for modular
forms by varying the modular forms in a p-adic family. This is a joint
work with Kiran Kedlaya and Jay Pottharst.