Colloquium

Manifold learning and an inverse problem for a wave equation

We consider invariant manifold learning and its applications in wave imaging. The invariant manifold learning problem, also known as the geometric Whitney problem, means the construction of a manifold $M$ and its Riemannian metric $g$ using a discrete metric space $(X,d_X)$ that approximates the manifold in the Gromov-Hausdorff sense. This problem is closely related to  manifold interpolation where a smooth $n$-dimensional surface $S\subset \mathbb R^m$, $m>n$ needs to be constructed to approximate a point cloud in $\mathbb R^m$. These questions are encountered in differential geometry, machine learning, and in many inverse problems encountered in applications. As an example, we consider an inverse problem for a wave equation $(\partial_t^2-\Delta_g)u(x,t)=F(x,t)$ on a Riemannian manifold $(M,g)$. We assume that we are given an open subset $V$ of $M$ and the source-to-solution map that maps a source supported in $V\times \mathbb R_+$ to the restriction of the solution $u$ in the set $V\times \mathbb R_+$. This map corresponds to the measurements made on the set $V$. The results on the first problem are done in collaboration with C. Fefferman, S. Ivanov, Y. Kurylev, and H. Narayanan, and the results on the second problem with R. Bosi and Y. Kurylev.

Positive metric entropy in nearly integrable Hamiltonian systems

The celebrated Kolmogorov-Arnold-Moser (KAM) theorem asserts that a small perturbation of an integrable Hamiltonian system preserves the quasi-periodicity of trajectories on a set of large measure. The question remains: how chaotic the system's behavior can be on the remaining "small" set? I will speak on a recent result of Dima Burago, Dong Chen and myself saying that every integrable system can be perturbed so that the resulting Hamiltonian system has positive measure-theoretic entropy.

Counting closed geodesics on hyperbolic surfaces II

Extreme Value Theory

Diophantine approximation: the classical, manifolds and Kleinian groups aspects

Mass transference principles and asymptotic properties of geodesics II

Mass transference principles and asymptotic properties of geodesics I