## Seminars and Colloquia by Series

Thursday, March 1, 2012 - 15:05 , Location: Skyles 006 , Gregorio Moreno Flores , University of Wisconsin, department of Mathematics , Organizer: Karim Lounici
The usual approach to KPZ is to study the scaling limit of particle systems. In this work, we show that the partition function of directed polymers (with a suitable boundary condition) converges, in a certain regime, to the Cole-Hopf solution of the KPZ equation in equilibrium. Coupled with some bounds on the fluctuations of directed polymers, this approach allows us to recover the cube root fluctuation bounds for KPZ in equilibrium. We also discuss some partial results for more general initial conditions.
Thursday, February 23, 2012 - 15:05 , Location: 006 , , Lehigh University , , Organizer: Ionel Popescu
We wish to understand ends of minimal surfaces contained in certain subsets of R^3. In particular, after explaining how the parabolicity and area growth of such minimal ends have been previously studied using universal superharmonic functions, we describe an alternative approach, yielding stronger results, based on studying Brownian motion on the surface. It turns out that the basic results also apply to a larger class of martingales than Brownian motion on a minimal surface, which both sheds light on the underlying geometry and potentially allows applications to other problems.
Thursday, February 16, 2012 - 15:05 , Location: Skyles 006 , Oren Louidor , UCLA , Organizer: Karim Lounici
We consider random walks on Z^d among nearest-neighbor random conductances which are i.i.d., positive, bounded uniformly from above but which can be arbitrarily close to zero. Our focus is on the detailed properties of the paths of the random walk conditioned to return back to the starting point after time 2n. We show that in the situations when the heat kernel exhibits subdiffusive behavior --- which is known to be possible in dimensions d \geq 4-- the walk gets trapped for time of order n in a small spatial region. This proves that the strategy used to infer subdiffusive lower bounds on the heat kernel in earlier studies of this problem is in fact dominant. In addition, we settle a conjecture on the maximal possible subdiffusive decay in four dimensions and prove that anomalous decay is a tail and thus zero-one event. Joint work with Marek Biskup, Alexander Vandenberg and Alexander Rozinov.
Thursday, February 9, 2012 - 15:05 , Location: Skyles 006 , Fabrice Baudoin , University of Purdue , Organizer: Karim Lounici
Let $\M$ be a smooth connected manifold endowed with a smooth measure $\mu$ and a smooth locally subelliptic diffusion operator $L$ which is symmetric with respect to $\mu$. We assume that $L$ satisfies a generalized curvature dimension inequality as introduced by Baudoin-Garofalo \cite{BG1}. Our goal is to discuss functional inequalities for $\mu$ like the Poincar\'e inequality, the log-Sobolev inequality or the Gaussian logarithmic isoperimetric inequality.
Thursday, January 26, 2012 - 15:05 , Location: Skiles 006 , Jon Hosking , IBM Research Division, T. J. Watson Research Center , Organizer: Liang Peng
L-moments are expectations of certain linear combinations of order statistics. They form the basis of a general theory which covers the summarization and description of theoretical probability distributions, the summarization and description of observed data samples, estimation of parameters and quantiles of probability distributions, and hypothesis tests for probability distributions. L-moments are in analogous to the conventional moments, but are more robust to outliers in the data and enable more secure inferences to be made from small samples about an underlying probability distribution. They can be used for estimation of parametric distributions, and can sometimes yield more efficient parameter estimates than the maximum-likelihood estimates. This talk gives a general summary of L-moment theory and methods, describes some applications ranging from environmental data analysis to financial risk management, and indicates some recent developments on nonparametric quantile estimation, "trimmed" L-moments for very heavy-tailed distributions, and L-moments for multivariate distributions.
Thursday, January 19, 2012 - 15:05 , Location: skyles 006 , Christophe Gomez , Department of Mathematics, Stanford University , Organizer: Karim Lounici
In this talk we will describe the different behaviors of solutions of the random Schrödinger with long-range correlations. While in the case of arandom potential with rapidly decaying correlations nontrivial phenomenaappear on the same scale, different phenomena appear on different scalesfor a random potential with slowly decaying correlations nontrivial .
Thursday, December 8, 2011 - 15:05 , Location: skyles 006 , Javier Rojo , Department of Statistics, Rice University , Organizer: Karim Lounici
We review various classifications of probability distributions based on their tail heaviness. Using a characterization of medium-tailed distributions we propose a test for testing the null hypothesis of medium-tail vs long- or short-tailed distributions. Some operating characteristics of the proposed test are discussed.
Thursday, November 10, 2011 - 15:05 , Location: skyles 006 , Ruodu Wang , School of mathematics, Georgia institute of Technology , Organizer: Karim Lounici
The marginal distribution of identically distributed random variables having a constant sum is called a completely mixable distribution. In this talk, the concept, history and present research of the complete mixability will be introduced. I will discuss its relevance to existing problems in the Frechet class, i.e. problems with known marginal distributions but unknown joint distribution and its applications in quantitative risk management.
Thursday, November 3, 2011 - 15:05 , Location: Skiles 006 , Alexey Shashkin , Moscow State University , Organizer:
Excursion sets of stationary random fields have attracted much attention in recent years.They have been applied to modeling complex geometrical structures in tomography, astro-physics and hydrodynamics. Given a random field and a specified level, it is natural to studygeometrical functionals of excursion sets considered in some bounded observation window.Main examples of such functionals are the volume, the surface area and the Euler charac-teristics. Starting from the classical Rice formula (1945), many results concerning calculationof moments of these geometrical functionals have been proven. There are much less resultsconcerning the asymptotic behavior (as the window size grows to infinity), as random variablesconsidered here depend non-smoothly on the realizations of the random field. In the talk wediscuss several recent achievements in this domain, concentrating on asymptotic normality andfunctional central limit theorems.
Thursday, October 27, 2011 - 15:05 , Location: Skyles 006 , , The Wharton School, Department of Statistics, University of Pennsylvania , , Organizer: Karim Lounici
We consider the problem of estimating the covariance matrix. Factormodels and random effect models have been shown to provide goodapproximations in modeling multivariate observations in many settings. These models motivate us to consider a general framework of covariancestructures, which contains sparse and low rank components. We propose aconvex optimization criterion, and the resulting estimator is shown torecover exactly the rank and support of the low rank and sparsecomponents respectively. The convergence rates are also presented. Tosolve the optimization problem, we propose an iterative algorithm basedon Nesterov's method, and it converges to the optimal with order 1/t2for any finite t iterations. Numerical performance is demonstratedusing simulated data and stock portfolio selection on S&P 100.(This is joint work with T. Tony Cai.)