Georgia Institute of Technology College of Sciences

We consider a model of randomly colliding particles interacting with a thermal bath. Collisions between particles are modeled via the Kac master equation while the thermostat is seen as an infinite gas at thermal equilibrium at inverse temperature \beta. The system admits the canonical distribution at inverse temperature \beta as the unique equilibrium state.

Lots of attention and research activity has been devoted to partially hyperbolic dynamical systems and their perturbations in the past few decades; however, the main emphasis has been on features such as stable ergodicity and accessibility rather than stronger statistical properties such as existence of SRB measures and exponential decay of correlations. In fact, these properties have been previously proved under some specific conditions (e.g. Anosov flows, skew products) which, in particular, do not persist under perturbations.

We study a gas of N hard disks in a box with semi-periodic boundary conditions. The unperturbed gas is hyperbolic and ergodic (these facts are proved for N=2 and expected to be true for all N>2). We study various perturbations by "twisting" the outgoing velocity at collisions with the walls. We show that the dynamics tends to collapse to various stable regimes, however we define the perturbations and however small they are.

In this talk I will discuss a family of lower bounds on the indirect Coulomb energy for atomic and molecular systems in two dimensions in terms of a functional of the single particle density with gradient correction terms

Consider an N by N matrix X of complex entries with iid real and imaginary parts with probability distribution h where h has Gaussian decay. We show that the local density of eigenvalues of X converges to the circular law with probability 1. More precisely, if we let a function f (z) have compact support in C and f_{\delta,z_0} (x) = f ( z-z^0 / \delta ) then the sequence of densities (1/N\delta^2) \int f_\delta d\mu_N converges to the circular law density (1/N\delta^2) \int f_\delta d\mu with probability 1. Here we show

We will discuss aspects of Chern-Simons theory, quantization and algebraic curves that appear in moduli spaces problems.

I'll talk about recent work, jointly with J. Baker, F. Klopp, S. Nakamura and G. Stolz concerning the random displacement model. I'll outline a proof of localization near the edge of the deterministic spectrum. Localization is meant in both senses, pure point spectrum with exponentially decaying eigenfunctions as well as dynamical localization. The proof relies on a well established multiscale analysis and the main problem is to verify the necessary ingredients, such as a Lifshitz tail estimate and a Wegner estimate.

This talk concerns aperiodic repetitive Delone sets and the dynamical systems associated with them. A typical example of an aperiodic repetitive Delone set is given by the set of vertices of the Penrose tiling. We show that natural questions concerning aperiodic repetitive Delone sets are reduced to the study of some cohomological equations on the associated dynamical systems. Using the formalism of tower systems introduced by Bellissard, Benedetti, and Gambaudo, we will study the problem about the existence of solution of these cohomological equations.

The McK--V system is a non--linear diffusion equation with a non--local non--linearity provided by convolution. Recently popular in a variety of applications, it enjoys an ancient heritage as a basis for understanding equilibrium and near equilibrium fluids. The model is discussed in finite volume where, on the basis of the physical considerations, the correct scaling (for the model itself) is identified. For dimension two and above and in large volume, the phase structure of the model is completely elucidated in (somewhat