## Nodal Lines of Random Waves

M. Sodin
Tel Aviv University
October 26, 2010

In the talk, I will describe recent attempts to understand the mysterious and beautiful geometry of nodal lines of random spherical harmonics and of random plane waves. If time permits, I will also discuss asymptotic statistical topology of other natural polynomial-like ensembles of random functions. The talk is based on a joint work with Fedja Nazarov.

## A Stationary Phase Method for a Class of Nonlinear Equations

Yen Do
Georgia Institute of Technology
October 26, 2010

In this talk I will describe a real-variable method to extract long-time asymptotics for solutions of many nonlinear equations (including the Schrodinger and mKdV equations). The method has many resemblances to the classical stationary phase method in the theory of oscillatory integrals.

## Lecture 3

Pierre Colmez
National Center for Scientific Research; Member, School of Mathematics
October 26, 2010

## A Semistable Model for the Tower of Modular Cures

Jared Weinstein
The usual Katz-Mazur model for the modular curve $X(p^n)$ has horribly singular reduction. For large n there isn't any model of $X(p^n)$ which has good reduction, but after extending the base one can at least find a semistable model, which means that the special fiber only has normal crossings as singularities. We will reveal a new picture of the special fiber of a semistable model of the entire tower of modular curves. We will also indicate why this problem is important from the point of view of the local Langlands correspondence for $GL(2)$ .