Seminars Sorted by Series

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...

We study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a...

I will discuss recent work on the global stability of the Euler-Maxwell equations in 3D (joint work with Guo and Pausader), and of the gravity water-wave system in 2D (joint work with Pusateri).

The talk is based on a recent work of M. Avdeeva (Princeton University), D. Li (IAS) and Ya. G. Sinai (Princeton University). We consider some new probability distributions related to the Mobius function and discuss their statistical properties. A...

I present a new, fully anisotropic, criterion for formation of trapped surfaces in vacuum obtained in collaboration with J. Luk and I. Rodnianski. We provide conditions on null data, concentrated in a neighborhood of a short null geodesic segment...

Stochastic quantization equations are evolutionary PDEs driven by space-time white noises. They are proposed by physicists in the 80s as the natural dynamics associated to the (Euclidean) quantum field theories. We will discuss the recent progress...

We discuss our results on global existence and convergence of solutions to the gradient flow equation for the Yang-Mills energy functional over a closed, four-dimensional, Riemannian manifolds: If the initial connection is close enough to a minimum...

Random band matrices (RBM) are natural intermediate models to study eigenvalue statistics and quantum propagation in disordered systems, since they interpolate between mean-field type Wigner matrices and random Schrodinger operators. In particular...

Numerous manifestations of wave localization permeate acoustics, quantum physics, mechanical and energy engineering. It was used in construction of noise abatement walls, LEDs, optical devices, to mention just a few applications. Yet, no systematic...

We will mainly report on the progress done recently the connectedness properties of the set of non-differentiable points of viscosity solutions of the Hamilton-Jacobi equation. To make the lecture accessible to people with no previous knowledge in...

I will discuss results on local eigenvalue statistics for uniform random regular graphs. For graphs whose degrees grow slowly with the number of vertices, we prove that the local semicircle law holds at the optimal scale, and that the bulk...

The goal of this talk is to explain universality for random band matrices, for band width comparable to the matrix size. Patching of quantum unique ergodicity on successive blocks plays a key role in proving random matrix statistics for such non...

The conventional perturbative approach and the nonperturbative lattice approach are the two standard yet very distinct formulations of quantum gauge theories. Since in dimension two Yang-Mills theory has a rigorous continuum limit of the lattice...

A spectral gap on a noncompact Riemannian manifold is an asymptotic strip free of resonances (poles of the meromorphic continuation of the resolvent of the Laplacian). The existence of such gap implies exponential decay of linear waves, modulo a...

We study the number of nodal domains of toral Laplace eigenfunctions. Following Nazarov-Sodin's results for random fields and Bourgain's de-randomisation procedure we establish a precise asymptotic result for "generic" eigenfunctions. Our main...

In this talk I will present a recent proof of a conjecture of C. Villani, namely the exponential convergence of solutions of spatially inhomogenous Boltzmann equations, with hard sphere potentials, to some equilibriums, called Maxwellians.

In this talk, I will discuss two problems concerning random data Cauchy theory for nonlinear wave equations. The first, based on joint work with Luhrmann, focuses on nonlinear wave equations with defocusing energy-subcritical power-type nonlinearity...

I will discuss the Kolmogorov equation, a simplest kinetic Fokker-Planck equation in the presence of boundaries. In the case of an absorbing boundary, I will present the well-posedness theory of classical solutions and Holder continuity of such...

A distinct covering system of congruences is a finite collection of arithmetic progressions to distinct moduli \[ a_i \bmod m_i, 1 m_1 m_2 \cdots m_k \] whose union is the integers. Answering a question of Erdős, I have shown that the least...

In 1979, O. Heilmann and E.H. Lieb introduced an interacting dimer model with the goal of proving the emergence of a nematic liquid crystal phase in it. In such a phase, dimers spontaneously align, but there is no long range translational order...

We present a new approach to the existence of time quasi-periodic solutions to nonlinear PDE's. It is based on the method of Anderson localization, harmonic analysis and algebraic analysis. This can be viewed as an infinite dimensional analogue of a...

We will describe an implementation of the Wiener theorem in $L^1$ type convolution algebras in the setting of spectral theory. In joint work with Marius Beceanu we obtained a structure theorem for the wave operators by this method.

I will consider the energy-critical wave maps equation with values in the sphere in the equivariant case, that is for symmetric initial data. It is known that if the initial data has small energy, then the corresponding solution scatters. Moreover...

We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water waves solutions, namely periodic and even in the space variable $x$, of a bi-dimensional ocean with finite depth under the...

I will discuss a connection between monodromy groups of variations of Hodge structure and the global behavior of the associated period map. The large-scale information in the period map is contained in the Lyapunov exponents, which are invariants...

For initial datum of finite kinetic energy Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this talk, I will discuss very recent joint work with Vlad Vicol in...

In spin systems, the existence of a spectral gap has far-reaching consequences. So-called "frustration-free" spin systems form a subclass that is special enough to make the spectral gap problem amenable and, at the same time, broad enough to include...

Given a measure preserving dynamical system, a real-valued observable determines a random process (by composing the observable with the iterates of the transformation). An important topic in ergodic theory is the study of the statistical properties...