# School of Mathematics

## C^0 Limits of Hamiltonian Paths and Spectral Invariants

After reviewing spectral invariants, I will write down an estimate, which under certain assumptions, relates the spectral invariants of a Hamiltonian to the C^{0}-distance of its flow from the identity. I will also show that, unlike the Hofer norm, the spectral norm is C^{0}-continuous on surfaces. Time permitting, I will present an application to the study of area preserving disk maps.

## Stochastic Twist Maps and Symplectic Diffusions

## A New Formulation of the Gross-Zagier Formula

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.

## Braids and Flows

## Arnold Diffusion by Variational Methods

## On the Rigidity of Black Holes

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 unrealistic analyticity assumption for the metric. The goal of the talk is to survey recent results, obtained in collaboration with Alexakis and Ionescu, on the general rigidity problem, without analyticity.

## The Mathematical Challenge of Large Networks

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; the world wide web, a network of webpages and hyperlinks). Various social networks, several of them created by the internet, are studied by sociologist, historians, epidemiologists, and economists. Huge networks arise in biology (from ecological networks to the brain), physics, and engineering.

## On the Instability for 2D Fluids

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.