Analysis Seminar

The Ginzburg-Landau theory was first developed to explain magnetic and other properties of superconductors, but had a profound influence on physics well beyond its original area. It had the first demonstration of the Higgs mechanism and it became a...

The Fermionic Fokker-Planck equation is a quantum-mechanical analog of the classical Fokker-Planck equation with which it has much in common, such as the same optimal hypercontractivity properties. In this paper we construct a Riemannian metric on...

I will report on recent joint work with L. Lanzani on three basic projection operators, each associated to an appropriate domain in C^n. These are: variants of Cauchy-Fantappie integrals; the Cauchy-Szego projection: and the Bergman projection. The...

As is well known, Eulerian simulations with a very small spatial mesh using an explicit scheme also require very small time steps, because the latter must be smaller than the time required to travel accross the mesh at the maximum flow velocity...

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...