Analysis Seminar

The Strauss conjecture for the Minkowski spacetime in three dimensions states that the semilinear equation \[\Box u=|u|^p,\ u(0) =\epsilon f,\ \partial_t u(0) = \epsilon g\] has a global solution for all $f$ and$g$ smooth, compactly supported and $...

We consider the non-local and non-linear equation $(-\Delta)^sQ+Q-Q^{\alpha+1}= 0$ involving the fractional Laplacian $(-\Delta)^s$ with $0 < s <1$. We prove uniqueness of energy minimizing solutions for the optimal range of $\alpha$'s. As a technical key result, we show that the associated linearized operator is nondegenerate, in the sense that its kernel is spanned by $\nabla Q$. This solves an open problem posed by Weinstein and by Kenig, Martel and Robbiano.

The talk is based on joint work with E. Lenzmann and L. Sylvestre.

We study the hole probability of Gaussian entire functions. More specifically, we work with entire functions given by a Taylor series with i.i.d complex Gaussian random variables and arbitrary non-random coefficients. A 'hole' is the event where the...

One of the principal questions about L-functions is the size of their critical values. In this talk, we will present a new subconvexity bound for the central value of a Dirichlet L-function of a character to a prime power modulus, which breaks a...

The interface between water and vacuum (governed by the "water wave equation"), and the interface between oil and water in sand (governed by the "Muskat equation") can develop singularities in finite time. Joint work with A. Castro, D. Cordoba, F...

We describe some results concerning the number of connected components of nodal lines of high frequency Maass forms on the modular surface. Based on heuristics connecting these to a critical percolation model, Bogomolny and Schmit have conjectured...

I shall describe several techniques for finding approximate solutions to the time-dependent Schr\"odinger equation in the semiclassical limit. The first of these involves expansions in "semiclassical wave packets" that are also sometimes called...

Consider the focusing semilinear wave equation in $R^3$ with energy-critical non-linearity \[ \partial_t^2 \psi - \Delta \psi - \psi^5 = 0,\ \psi(0) = \psi_0,\ \partial_t \psi(0) = \psi_1. \]

This equation admits stationary solutions of the form \[...

We shall discuss reducibility of these equations on the torus with a small potential that depends quasi-periodically on time. Reducibility amounts to "reduce” the equation to a time-independent linear equation with pure point spectrum in which case...