Georgia Institute of Technology College of Sciences

I will talk about two model problem concerning a diffusion with a cellular drift (a.k.a array of opposing vortices). The first concerns the expected exit time from a domain as both the flow amplitude $A$ (or more precisely the Peclet number) goes to infinity, AND the cell size (or vortex seperation) $\epsilon$ approaches $0$ simultaneously. When one of the parameters is fixed, the problem has been extensively studied and the limiting behaviour is that of an effective "homogenized" or "averaged" problem. When both vary simultaneously one sees an interesting transition at $A \approx \eps^{-4}$. While the behaviour in the averaged regime ($A \gg \eps^{-4}$) is well understood, the behaviour in the homogenized regime ($A \ll \eps^{-4}$) is poorly understood, and the critical transition regime is not understood at all. The second problem concerns an anomalous diffusive behaviour observed in "intermediate" time scales. It is well known that a passive tracer diffusing in the presence of a strong cellular flows "homogenizes" and behaves like an effective Brownian motion on large time scales. On intermediate time scales, however, an anomalous diffusive behaviour was numerically observed recently. I will show a few preliminary rigorous results indicating that the stable "anomalous" behaviour at intermediate time scales is better modelled through Levy flights, and show how this can be used to recover the homogenized Brownian behaviour on long time scales.

In this talk I will describe a theory of matrix completion for the extreme case of noisy 1-bit observations. In this setting, instead of observing a subset of the real-valued entries of a matrix M, we obtain a small number of binary (1-bit) measurements generated according to a probability distribution determined by the real-valued entries of M. The central question I will address is whether or not it is possible to obtain an accurate estimate of M from this data. In general this would seem impossible, but I will show that the maximum likelihood estimate under a suitable constraint returns an accurate estimate of M when $\|M\|_{\infty} \le \alpha$ and $\rank(M) \le r$. If the log-likelihood is a concave function (e.g., the logistic or probit observation models), then we can obtain this maximum likelihood estimate by optimizing a convex program. I will also provide lower bounds showing that this estimate is near-optimal and illustrate the potential of this method with some preliminary numerical simulations.

In small dimension a random geometric graph behaves very differently from a standard Erdös-Rényi random graph. On the other hand when the dimension tends to infinity (with the number of vertices being fixed) both models coincides. In this talk we study the behavior of the clique number of random geometric graphs when the dimension grows with the number of vertices.

There is a long history on the study of zeros of random polynomials whose coefficients are independent, identically distributed, non-degenerate random variables. We will first provide an overview on zeros of random functions and then show exact and/or asymptotic bounds on probabilities that all zeros of a random polynomial are real under various distributions. The talk is accessible to undergraduate and graduate students in any areas of mathematics.