Georgia Institute of Technology College of Sciences

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.

Random and irregular growth is all around us. We see it in the form of cancer growth, bacterial infection, fluid flow through porous rock, and propagating flame fronts. In this talk, I will introduce several different models for random growth and the different shapes that can arise from them. Then I will talk in more detail about one model, first-passage percolation, and some of the main questions that researchers study about it.

Quantum graphs are metric graphs with differential equations defined on the edges. Recent interest in control and inverse problems for quantum graphs
is motivated by applications to important problems of classical and quantum physics, chemistry, biology, and engineering.

In this talk we describe some new controllability and identifability results for partial differential equations on compact graphs. In particular, we consider graph-like networks of inhomogeneous strings with masses attached at the interior vertices. We show that the wave transmitted through a mass is more
regular than the incoming wave. Therefore, the regularity of the solution to the initial boundary value problem on an edge depends on the combinatorial distance of this edge from the source, that makes control and inverse problems
for such systems more diffcult.

We prove the exact controllability of the systems with the optimal number of controls and propose an algorithm recovering the unknown densities of thestrings, lengths of the edges, attached masses, and the topology of the graph. The proofs are based on the boundary control and leaf peeling methods developed in our previous papers. The boundary control method is a powerful
method in inverse theory which uses deep connections between controllability and identifability of distributed parameter systems and lends itself to straight-forward algorithmic implementations.