Seminars Sorted by Series

Let p_1,...,p_k be integer polynomials of one or several variables. There is a relation between the density of polynomial configurations a+p_1(n),...,a+p_k(n) in sets of integers and the form of the closure of the diagonal of X^k under the...

In this talk I will consider actions of non-abelian groups on n-dimensional tori, explain the notions of stiffness and stationary measures, and show how under fairly general assumptions stationary measures can be classified. A key ingredient is a...

In 1975 Szemeredi proved that every subset of the integers with positive density contains arbitrarily long arithmetic progressions. Bergelson and Leibman showed in 1996 that the common difference of the arithmetic progression can be a square, a cube...

We investigate the Bilinear Hilbert Transform in the plane and the pointwise convergence of bilinear averages in Ergodic theory, arising from Z^2 actions. Our techniques combine novel one and a half dimensional phase-space analysis with more...

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.

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...

In this talk, we will present some recent results in the study of the nonlinear wave equation with cubic defocusing nonlinearity, describing the completion of a program to establish global well-posedness and scattering in the energy-supercritical...

The Davenport-Heilbronn function (introduced by Titchmarsh) is a linear combination of the two L-functions with a complex character mod 5, with a functional equation of L-function type but for which the analogue of the Riemann hypothesis fails. In...

Calibrated currents are a particular class of volume-minimizers and as such provide interesting explicit examples of solutions to Plateau's problem. Their role goes however much beyond that: they naturally appear when dealing with several geometric...

I will discuss some recent work, joint with B. Pausader, on constructing global solutions of defocusing energy-critical nonlinear Schrodinger equations in periodic and semiperiodic settings.

I shall explain the structure of correlation functions for square-free numbers and describe a 'natural' dynamical system associated to them. Spectral analysis allows us to show that this system is metrically isomorphic to a translation on a compact...

Consider the flow of ideal incompressible fluid in a bounded 2-d domain $M$ (say, $M= 3DT^2$, the 2-d torus). In the Lagrange formulation, the flow is a geodesic $f_t$ on the group $SDif f(M)$ of volume-preserving diffeomorphisms of $M$ with respect...

Zaremba's 1971 conjecture predicts that every integer appears as the denominator of a finite continued fraction whose partial quotients are bounded by an absolute constant. We confirm this conjecture for a set of density one.

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...

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 \[...

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...

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...

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...

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...

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...

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.

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 $...

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...

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...

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...

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...

We study the nonlinear Klein-Gordon equation, in one dimension, with a qudratic term and variable coefficient qubic term. This equation arises from the asymptotic stability theory of the kink solution.Our main result is the global existence and...

I present some new dispersive estimates for Schroedinger's equation with a time-dependent potential, together with applications.

Several examples of Hamiltonian evolution equations for systems with infinitely many degrees of freedom are presented. It is sketched how these equations can be derived from some underlying quantum dynamics ("mean-field limit") and what kind of...

We develop a simple geometric variant of the Kabatiansky-Levenshtein approach to proving sphere packing density bounds. This variant gives a small improvement to the best bounds known in Euclidean space (from 1978) and an exponential improvement in...

This is the second of two talks in which the speaker will discuss the development of the theory of Toeplitz matrices and determinants in response to questions arising in the analysis of the Ising model of statistical mechanics. The first talk will...

We will discuss recent work on wave evolutions for large data. Particular emphasis will be placed on concentration compactness ideas. Amongst others, we will describe a result for wave equations from R^3 minus the unit ball into the sphere S^3 where...

I will present new results concerning the approximation of the BV-norm by nonlocal, nonconvex, functionals. The original motivation comes from Image Processing. Numerous problems remain open. The talk is based on a joint work with H.-M. Nguyen.

In this talk, we present recent works with Jean Bourgain on global well-posedness for the radial nonlinear Schr\"odinger and wave equations set on the unit ball in $\mathbb{R}^N$ with supercritical data chosen randomly in the support of the...

I will discuss some recent results on partial regularity of solutions to the 4D non-stationary Navier-Stokes equations and the 6D stationary Navier-Stokes equations.

Resonances are complex analogs of eigenvalues for Laplacians on noncompact manifolds, arising in long time resonance expansions of linear waves. We prove a Weyl type asymptotic formula for the number of resonances in a strip, provided that the set...

Calibrated currents naturally appear when dealing with several geometric questions, some aspects of which require a deep understanding of regularity properties of calibrated currents. We will review some of these issues, then focusing on the two...

We present some novel approaches to the instability problem of Hamiltonian systems (in particular, the Arnold Diffusion problem). We show that, under generic conditions, perturbations of geodesic flows by recurrent dynamics yield trajectories whose...