Georgia Institute of Technology College of Sciences

Let (A_n) be a sequence of random matrices, such that for every n, A_n is n by n with i.i.d. entries, and each entry is of the form b*x, where b is a Bernoulli random variable with probability of success p_n, and x is an independent random variable of unit variance. We show that, as long as n*p_n converges to infinity, the appropriately rescaled spectral distribution of A_n converges to the uniform measure on the unit disc of complex plane. Based on joint work with Mark Rudelson.

This is an interdisciplinary event using puzzles, story-telling, and original music and dance to interpret Euler's analysis of the problem of the Seven Bridges of Königsberg, and the birth of graph theory. Beginning at 11:00, students from GT's Club Math will be on the plaza between the Howie and Mason Buildings along Atlantic Dr., with information and hands-on puzzles related to Euler and to graphs. At 12:00 the performance will begin, as the GT Symphony Orchestra and a team of dancers interpret the story of the Seven Bridges. For more information see the news article at http://hg.gatech.edu/node/610095.

We investigate a dynamical system consisting of $N$ particles moving on a $d$-dimensional torus under the action of an electric field $E$ with a Gaussian thermostat to keep the total energy constant. The particles are also subject to stochastic collisions which randomize direction but do not change the speed. We prove that in the van Hove scaling limit, $E\to 0$ and $t\to t/E^2$, the trajectory of the speeds $v_i$ is described by a stochastic differential equation corresponding to diffusion on a constant energy sphere.Our results are based on splitting the system's evolution into a ``slow'' process and an independent ``noise''. We show that the noise, suitably rescaled, converges to a Brownian motion. Then we employ the Ito-Lyons continuity theorem to identify the limit of the slow process.

Koldobsky showed that for an arbitrary measure on R^n, the measure of the largest section of a symmetric convex body can be estimated from below by 1/sqrt{n}, in with the appropriate scaling. He conjectured that a much better result must hold, however it was recemtly shown by Koldobsky and Klartag that 1/sqrt{n} is best possible, up to a logarithmic error. In this talk we will discuss how to remove the said logarithmic error and obtain the sharp estimate from below for Koldobsky's slicing problem. The method shall be based on a "random rounding" method of discretizing the unit sphere. Further, this method may be effectively applied to estimating the smallest singular value of random matrices under minimal assumptions; a brief outline shall be mentioned (but most of it shall be saved for another talk). This is a joint work with Bo'az Klartag.