# School of Mathematics

## 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.

## Hodge Structures in Symplectic Geometry

## On the Long-Time Behavior of 2-D Flows

## Riemannian Exponential Map on the Group of Volume-Preserving Diffeomorphisms

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 \(L^2\) metric. I will describe some recent work on the structure of singularities of the associated exponential map and related results

## Rigidity of 3-Colorings of the d-Dimensional Discrete Torus

## The Universal Relation Between Exponents in First-Passage Percolation

It has been conjectured in numerous physics papers that in ordinary first-passage percolation on integer lattices, the fluctuation exponent \chi and the wandering exponent \xi are related through the universal relation \chi=2\xi -1, irrespective of the dimension. This is sometimes called the KPZ relation between the two exponents. I will give a rigorous proof of this conjecture assuming that the exponents exist in a certain sense.