## Seminars and Colloquia by Series

Thursday, February 28, 2013 - 15:05 , Location: Skiles 006 , Gautam Iyer , Carnegie Mellon , Organizer:
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.
Tuesday, February 19, 2013 - 15:05 , Location: Skyles 005 , Jon A. Wellner , University of Washington , Organizer: Karim Lounici
I will review recent progress concerning nonparametric estimation of log-concave densities and related families in $R^1$ and $R^d$.  In the case of $R^1$, I will present limit theory for the estimators at fixed points at which the population density has a non-zero second derivative and for the resulting natural mode estimator under a corresponding hypothesis. In the case of $R^d$ with $d\ge 2$  will briefly discuss some recent progress and sketch a variety of open problems.
Thursday, February 7, 2013 - 15:05 , Location: Skyles 006 , Mark Davenport , Georgia Institute of Technology , Organizer: Karim Lounici
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.
Thursday, January 31, 2013 - 15:05 , Location: Skiles 006 , Geordie Richards , IMA , Organizer:
The periodic generalized Korteweg-de Vries equation (gKdV) is a canonical dispersive partial differential equation with numerous applications in physics and engineering.  In this talk we present invariance of the Gibbs measure under the flow of the gauge transformed periodic quartic gKdV.  The proof relies on probabilistic arguments which exhibit nonlinear smoothing when the initial data are randomized.  As a corollary we obtain almost sure global well-posedness for the (ungauged) quartic gKdV at regularities where this PDE is deterministically ill-posed.
Thursday, January 24, 2013 - 15:05 , Location: Skyles 006 , Sebastien Bubeck , Princeton University , Organizer: Karim Lounici
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.
Thursday, January 17, 2013 - 15:05 , Location: Skiles 006 , Louis-Pierre Arguin , Université de Montréal , Organizer:
Gaussian fields with logarithmically decaying correlations, such as branching Brownian motion and the 2D Gaussian free field, are conjectured to form a new universality class of extreme value statistics (notably in the work of Carpentier & Ledoussal and Fyodorov & Bouchaud). This class is the borderline case between the class of IID random variables, and models where correlations start to affect the statistics. In this talk, I will report on the recent rigorous progress in describing the new features of this class. In particular, I will describe the emergence of Poisson-Dirichlet statistics. This is joint work with Olivier Zindy.
Thursday, December 6, 2012 - 15:05 , Location: Skiles 006 , Wenbo Li , University of Delaware , Organizer:
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.
Thursday, November 29, 2012 - 15:05 , Location: Skiles 006 , Tai Melcher , University of Virginia , Organizer:
Smoothness is a fundamental principle in the study of measures on infinite-dimensional spaces, where an obvious obstruction to overcome is the lack of an infinite-dimensional Lebesgue or volume measure. Canonical examples of smooth measures include those induced by a Brownian motion, both its end point distribution and as a real-valued path. More generally, any Gaussian measure on a Banach space is smooth. Heat kernel measure is the law of a Brownian motion on a curved space, and as such is the natural analogue of Gaussian measure there. We will discuss some recent smoothness results for these measures on certain classes of infinite-dimensional groups, including in some degenerate settings. This is joint work with Fabrice Baudoin, Daniel Dobbs, and Masha Gordina.
Thursday, November 15, 2012 - 15:05 , Location: Skyles 006 , Cun-Hui Zhang , Rutgers University , Organizer: Karim Lounici
This paper concerns the problem of matrix completion, which is to estimate a matrix from observations in a small subset of indices. We propose a calibrated spectrum elastic net method with a sum of the nuclear and Frobenius penalties and develop an iterative algorithm to solve the convex minimization problem. The iterative algorithm alternates between imputing the missing entries in the incomplete matrix by the current guess and estimating the matrix by a scaled soft-thresholding singular value decomposition of the imputed matrix until the resulting matrix converges. A calibration step follows to correct the bias caused by the Frobenius penalty. Under proper coherence conditions and for suitable penalties levels, we prove that the proposed estimator achieves an error bound of nearly optimal order and in proportion to the noise level. This provides a unified analysis of the noisy and noiseless matrix completion problems. Tingni Sun and Cun-Hui Zhang, Rutgers University
Thursday, November 8, 2012 - 15:05 , Location: Skiles 006 , Chris Evans , University of Missouri , Organizer:
In a series of famous papers E. Wong and M. Zakai showed that the solution to a Stratonovich SDE is the limit of the solutions to a corresponding ODE driven by the piecewise-linear interpolation of the driving Brownian motion. In particular, this implies that solutions to Stratonovich SDE "behave as we would expect from ODE theory". Working with my PhD adviser, Daniel Stroock, we have shown that a similar approximation result holds, in the sense of weak convergence of distributions, for reflected Stratonovich SDE.