Seminars Sorted by Series

We show that a random simple curve in a planar n-connected domain that is conformally invariant and satisfies a Markovian-type property, can be described by a diffusion on a moduli space of dimension 3n-2. Under a natural symmetry condition...

"We give an overview of the ideas and techniques relating these seemingly different subjects. I will start from the classical examples, such as enumeration of triangulations by means of one matrix model and counting of colored graphs (Ising model on...

This will be an overview of the paper hep-th/0306238 written jointly with N. Nekrasov. Our main idea is the interpretation of the low-energy effective prepotential of the N=2 supersymmetric gauge theory as the free energy of a certain natural...

A (one-dimensional) translation invariant point process of bounded variability is one in which the variance of the number of particles in any interval is bounded, uniformly in the length of the interval. This represents a strong suppression of...

We have proposed that the glass transition is one example of a broader class of jamming transitions, where systems can develop extremely long stress relaxation times in disordered states as temperature is lowered, an applied shear stress is lowered...

We will discuss this model of a random simple path and its connection to spanning trees, matrix formulas, the Potts model and SLE. Time permitting, we shall discuss the proof the it has a scaling limit in three dimensions. No prior knowledge will be...

We construct an ensemble of (sparse) random matrices whose eigenvalues follow the Gibbs distribution for n particles of Coulomb gas on the unit circle at any given inverse temperature. Our approach combines elements from the theory of orthogonal...

This is joint work with Dimitri Gioev. The speaker will show how to prove universality in the bulk and at the edge for orthogonal and symplectic ensembles of random matrices with weights of the form exp(-V(x))dx. The method follows the formalism of...

Within non-relativistic quantum electrodnamics, atoms interacting with the radiation field are expected to have a ground state. It is further expected that the ground state exists independently of the size of the coupling constant $\alpha$ and the...

The integer quantum Hall effect (IQHE) entails a very precise quantization of the Hall conductance in a 2D sample at very low temperatures. Depending on whether the currents in the sample are ascribed to the bulk or the edge, two apparently...

We consider a gas of fermions with non-zero spin at positive temperature $T$. We show that if the range of the interparticle interaction is small compared to the mean particle distance, the thermodynamic pressure differs to leading order from the...

We discuss the problem of deriving Fourier's law of heat transport in a Hamiltonian system of coupled anharmonic oscillators subject to boundary noise and derive it in a closure approximation of the stationary state of the system.

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...