Video Lectures

After reviewing spectral invariants, I will write down an estimate, which under certain assumptions, relates the spectral invariants of a Hamiltonian to the C0-distance of its flow from the identity. I will also show that, unlike the Hofer norm, the...

A knot is more or less what you think it is—a tangled mess of string in ordinary three-dimensional space. In the twentieth century, mathematicians developed a rich and deep theory of knots. And surprisingly, as Edward Witten, Charles Simonyi...

In this talk, I will present a formulation of the Gross-Zagier formula over Shimura curves using automorphic representations with algebraic coefficients. It is a joint work with Shou-wu Zhang and Wei Zhang.

The classical result on the uniqueness of black holes in GR, due to Hawking, which asserts that regular, stationary solutions of the Einstein vacuum equations must be isometric to an admissible black hole Kerr solution, has at its core a a highly...

It is becoming more and more clear that many of the most exciting structures of our world can be described as large networks. The internet is perhaps the foremost example, modeled by different networks (the physical internet, a network of devices...

I will explain how essential information about the structure of symplectic manifolds is captured by algebraic data, and specifically by the non-commutative (mixed) Hodge structure on the cohomology of the Fukaya category. I will discuss how mirror...

For 2D Euler equation, we prove a double exponential lower bound on the vorticity gradient. We will also discus some further results on the singularity formation for other models.

In 1966 V. Arnold showed how solutions of the Euler equations of hydrodynamics can be viewed as geodesics in the group of volume-preserving diffeomorphisms. This provided a motivation to study the geometry of this group equipped with the \(L^2\)...