Seminars Sorted by Series

We examine several non convex free energy functionals involving a double well potential, and an energy term that penalizes variation in the mass density field. The simplest example is the so--called Cahn--Hilliard functional, which is purely...

I will present results on the critical behavior of directed polymer models interacting with a defect line, in presence of quenched disorder. These models show a localization-delocalization phase transition. Our main result is that the transition in...

A model based on independent bond percolation is defined. It is demonstrated that this model exhibits critical behavior and, at least at the level of Cardy's formula, has the same limiting continuum behavior as the site model.

I conjecture a form for the scaling limit of the partition function of the critical O(n) and Potts models on the annulus, using Coulomb gas methods. This has several subtleties whose elucidation sheds light on the nature of the Coulomb gas mapping.

Random band matrices have been proposed as an effective, or toy, model for a disorder induced localization-delocalization transition of eigenstates. Most results about these matrices, and the transition, are based on numerics or on calculations with...

This is continuation of the previous seminar in which the formulation of the renormalisation group is given in more detail: a space of statistical mechanical models is defined. The renormalisation group is a map on this space and there is a basic...

In this talk we introduce a new class of random fractals which we call conformal snowflakes. We study fine structure of harmonic measure on theses snowflakes. It turns out that in this case the multifractal spectrum of harmonic measure is related to...

Since the end of the nineties, the relations of optimal transport with many functional inequalities with geometric content has been revealed and explored by several authors (Barthe, Caffarelli, Cordero, McCann, Otto and others). Sobolev inequalities...

We will present some recent developments in the quasi-geostrophic equations. We show that local solutions to critical and super-critical dissipative quasi-geostrophic equations have higher regularity, although one gets lower derivative in the...

A metal or ceramic is naturally decomposed into cells called "grains". The geometry of this cell complex influences the properties of the material. Some interesting mathematical problems arise in trying to understand the time evolution of these...

Using the hydrodynamic limit theory, we will review the large deviations of the heat current through a diffusive system maintained off equilibrium by two heat baths at unequal temperatures. We will also discuss a toy model for dissipative dynamics...

Spectral prosperities of random matrices can be described in terms of correlations of a statistical mechanics model with hyperbolic symmetry. This talk will describe and analyze a simpler version of this model which is closely related to random walk...

Rigorous results about the classical microcanonical ensemble have so far been based on the regularization of the microcanonical measure, with the exception of the ideal gas. In this talk I explain that the regularization is not needed for...

We study the integrability properties of the gain part of the Boltzmann collision operator using radial symmetrization techniques from harmonic analysis to show Young's inequality for the case of hard potentials and the Hardy-Littlewood-Sobolev...

This talk will discuss non-spectral poles of the scattering operator arising in problems with regular singular point behavior at infinity. The phenomenon will be exhibited in a one-dimensional model example involving the Yukawa potential on a half...

The talk will consist of two parts. At first, we shall review connections between Lyapunov exponents, large deviations for diffusions in random media, and the homogenization problem for some stochastic Hamilton-Jacobi-Bellman equations. The second...

The Keller-Segal equation exhibits a competition between diffusion and effects leading to concentration, and depending on whether the total mass is above or below a critical value, one or the other wins. We examine the long time behavior for...

The Balescu-Lenard equation from kinetic theory--which was earlier derived in a more complicated form by Bogoliubov--is widely considered to include a highly accurate correction to Landau’s fundamental model for grazing collisions. So far this...

In this talk I will review two recent works in collaboration with A. Bertozzi and T. Laurent and with M. DiFrancesco, A. Figalli, T. Laurent and D. Slepcev in which we prove infinite/finite time blow-up for generic initial data in several models...

The equilibrium shape of a crystal is determined by the minimization under a volume constraint of its free energy, consisting of an anisotropic interfacial surface energy plus a bulk potential energy. In the absence of the potential term, the...

In this talk I review recent results on the infrared mass renormalization in non-relativistic quantum electrodynamics (non-relativistic particle system interacting with quantized electro-magnetic field). The review will also include the description...

The Schroedinger (and Landau-Lifshitz) map equations are a basic model in ferromagnetism, and a natural geometric (hence nonlinear) version of the Schroedinger (and Schroedinger-heat) equation. While there has been recent progress on the question of...

This talk is about renormalization group methods for quantum field theory and classical statistical mechanics. The renormalization group is not a group but a technique for reducing infinite dimensional integrals with strong correlations to a...

A corollary of a celebrated theorem of Nash and Kuiper is the existence of $C^1$ isometric embeddings of the standard 2-sphere in arbitrarily small balls of the euclidean 3-d space. On the contrary, the image of $C^2$ isometric embeddings are all...

We discuss the global regularity vs. finite time breakdown in nonlinear convection driven by different models of forcing. Finite time breakdown depends on whether the initial configuration crosses intrinsic, O(1) critical thresholds (CT). Our...

We discuss the dynamics of soliton-like solutions of the nonlinear Schroedinger-Gross Pitaevskii equation. After a review of basic results, we outline recent work on the large time energy distribution in multimoded systems and on gap-solitons...

We consider an open quantum system consisting of two spins 1/2 (qubits) interacting with thermal reservoirs (environments). Each spin is coupled to its own local reservoir, and the spins are coupled to a common third reservoir (collective coupling)...

In the '70s, Kadanoff, Luther and Peschel conjectured universal formulas among the critical indices of certain Ising-like, 2 dimensional, statistical models. We present a proof of some of these formulas.

The effect of interparticle interactions on the critical temperature for Bose-Einstein condensation has been a controversial issue in the physics literature. Various approximation schemes lead to different conclusions, concerning both the sign and...

I will present results on Glauber dynamics of Ising models and continuum
$\varphi^4$ measures.

For ferromagnetic Ising models, we show that the log-Sobolev constant satisfies a simple bound expressed only in terms of the susceptibility of the model...

I will discuss new constraints on the spectra of Maass forms on compact hyperbolic 2-orbifolds. The constraints arise from integrals of products of four functions in discrete series representations realized in $L^2(\Gamma\backslash G)$, where $...

Grassmann integrals are integrals over functions on an exterior algebra. In the "Gaussian" case they can be expressed as a determinant or Pfaffian.

These integral arise in two dimensional Ising models and random tilings or dimers. 

We review the...

In this talk we will describe the behaviors of flat surfaces and geodesics on hyperbolic surfaces, as their genera tend to infinity. We first discuss enumerative results that count the number of such surfaces or geodesics (which can be viewed as...

In this talk, we prove the invariance of the Gibbs measure for the three-dimensional cubic nonlinear wave equation, which is also known as the hyperbolic $\Phi ^{4} _{3}$-model.

In the first half of this talk, we illustrate our main objects and...

The question of optimizing an eigenvalue of a family of self-adjoint operators that depends on a set of parameters arises in diverse areas of mathematical physics.  Among the particular motivations for this talk are the Floquet-Bloch decomposition...

Liouville field theory was introduced by Polyakov in the eighties in the context of string theory. Liouville theory appeared there under the form of a 2D Feynman path integral, which can be thought of as a measure (or expectation value) over the...

Motivated by questions in computer science, we consider the problem of approximating local points (real or p-adic points) on the unit sphere S^d optimally by the projection of the integral points lying on R*S^d, where R^2 is an integer. We present...

We consider the problem of bounding the sup-norm of $L^2$-normalised Hecke-Laplace eigenforms $\phi_j$ on $S^3$. Along the way, we overcome the difficulty of possibly small eigenvalues in the Iwaniec-Sarnak amplifier by taking a whole space of...

In their seminal works from the 80's, Lubotzky, Phillips and Sarnak proved the following two results: (i) An explicit construction of Ramanujan regular graphs. (ii) An explicit method of placing points on the sphere uniformly equidistributed. These...