# Seminars & Events for 2008-2009

##### The Rudnick-Sarnak Conjectures

We'll discuss the quantum unique ergodicity conjecture of Rudnick and Sarnak and its holomorphic analogue. Highlighting the key ideas in recent joint with with K. Soundararajan, we'll demonstrate how one may use number theoretic techniques to solve this problem.

##### Parallel-in-time algorithms and long-time integration

We investigate some issues related to the integration of Hamiltonian systems when using integrators that are parallel in time (the so-called class of parareal integrators, introduced by JL Lions, Y. Maday and G. Turinici in C. R. Acad. Sci., Paris, Sér. I, Math. 332, No.7, 661-668 (2001)).

##### The Einstein-Weyl Equations, Scattering Maps, and Holomorphic Disks

This talk will show that conformally compact, globally hyperbolic, Lorentzian Einstein-Weyl 3-manifolds are in natural one-to-one correspondence with orientation-reversing diffeomorphisms of the 2-sphere. The proof hinges on a holomorphic-disk analog of Hitchin's mini-twistor correspondence.

##### p-adically completed cohomology and the p-adic Langlands program

Speaking at a general level, a major goal of the p-adic Langlands program (from a global, rather than local, perspective) is to find a p-adic generalization of the notion of automorphic eigenform, the hope being that every p-adic global Galois representation will correspond to such an object. (Recall that only those Galois representations that are motivic, i.e.

##### Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations

Consider axisymmetric strong solutions, v, of the incompressible Navier-Stokes equations in $\mathbb{R}^3$ with non-trivial swirl. Leray, in his 1934 Acta.

##### The Empirical Mode Decomposition: the method, its progress, and open questions

The Empirical Mode Decomposition (EMD) was an empirical one-dimensional data decomposition method invented by Dr. Norden Huang about ten years ago and has been used with great success in many fields of science and engineering.

##### On the evolution of solutions to a many-body Schrödinger equation

In part I, I will describe background material and a new proof for the uniqueness of solutions to the Gross-Pitaevskii hierarchy. This is joint work with S. Klainerman and is a simplification, based on space-time estimates, of an older proof of Erdös, Schlein and Yau.

##### What goes on in a plasma

##### Sum-product estimates via combinatorial geometry

Every two-dimensional drawing of any graph with $V$ vertices and $E\ge 4V$ edges necessarily has at least $E3/V2$ pairs of crossing edges. Also, for every set $A$ of real numbers, one of $A+A$ (the set of all pairwise sums of elements of $A$) or $A\cdot A$ (the set of all pairwise products) has size at least $|A|5/4$.

##### Local limit theorems in ergodic theory

We use Stone's version of a local limit theorem from 1969: Let $(X,{\cal F},T,m)$ be a measure preserving dynamical system.

##### Avoiding small subgraphs in Achlioptas processes

Consider the following random process. At each round, one is presented with two random edges from the edge set of the complete graph on n vertices, and is asked to choose one of them. The selected edges are collected into a graph, which thus grows at the rate of one edge per round.

##### Recurrence of random paths and counting closed geodesics in strata

We discuss the problem of counting closed geodesics in a stratum of the moduli space of Abelian(quadratic) differentials. This is joint work with Alex Eskin and Kasra Rafi.

##### CM liftings of abelian varieties

##### On the structure of Lagrangian submanifolds

This is a report on a recent joint project with Lars Schaefer. We derive results related to the minimality of Lagrangian submanifolds. In particular, these apply to Lagrangian 3-folds and to Lagrangian submanifolds in twistor spaces over quaternionic Kaehler manifolds. We then use a splitting theorem to give a better description in dimensions four and five.

##### On the interplay between coding theory and compressed sensing

Compressed sensing (CS) is a signal processing technique that allows for accurate, polynomial time recovery of sparse data-vectors based on a small number of linear measurements.

##### Landau damping

Sixty years ago, Landau discovered a paradoxical collisionless relaxation effect in plasmas. The Landau damping is now one of the cornerstones of classical plasma physics. From the mathematical point of view, it has remained elusive so far, since the best available results prove the existence of some damped solutions, without saying anything about their genericity.

##### Warped Convolutions: A novel tool in the construction of quantum field theories

Recently, Grosse and Lechner introduced a deformation procedure for non-interacting quantum field theories, giving rise to interesting examples of theories with non-trivial scattering matrix in any number of spacetime dimensions.

##### Arakelov invariants on modular curves

Arakelov theory provides a rich set of invariants. We shall discuss the question of their limiting behavior in several classical examples, with an emphasis on heights of special points and of modular curves.

##### On a conjecture of De Giorgi

In 1978 De Giorgi made a conjecture about the symmetry of global solutions to a certain semilinear elliptic equation. He stated that monotone, bounded solutions of $$ \triangle u=u^3-u$$ in $\mathbb{R}^n$ are one dimensional (i.e. the level sets of $u$ are hyperplanes) at least in dimension $n \le 8$.

##### Poisson summation formula

I will first talk about the classical Poisson summation formula and then about a vast generalization of it, namely the trace formula.