ColumbiaPrinceton Probability Day 2013
TechTalks from event: ColumbiaPrinceton Probability Day 2013

About Heavy Tails Random MatricesWigner's matrices are Hermitian matrices with independent entries modulo the symmetry constraint. In the last few years, it was shown that the properties of such matrices are similar to those of a matrix with Gaussian entries propiding the entries have enough finite moments. In this talk we will investigate the properties of matrices which do not belong to the universality class of Wigner matrices because their entries have heavy tails.

Heat Flow, Harnack Inequalities, and Optimal TransportationThe talk will develop connections between Harnack inequalities for the heat flow of diffusion operators with curvature bounded from below and optimal transportation. Through heat kernel inequalities, a new isoperimetrictype Harnack inequality is emphasized. Commutation properties between the heat and HopfLax semigroups are developed consequently, providing direct access to the heat flow contraction property along Wasserstein distances.

Robust Optimality of Gaussian Noise StabilityIn 1985 C. Borell proved that under the Gaussian measure, halfspaces are the most stable sets. While a number of proofs of this result were discovered over the years, it was not known if halfspaces are the unique optimizers. The talk will survey recent results with Joe Neeman establishing that halfspaces are uniquely the most noise stable sets. Furthermore, we prove a quantitative dimension independent versions of uniqueness, showing that a set which is almost optimally noise stable must be close to a halfspace. Our work answers a question of Ledoux from 1994 and has numerous applications in theoretical computer science and social choice.

Geometric Applications of Markov ChainsThe deep and fruitful interactions between probability and geometry are wellestablished, including the powerful use of probabilistic constructions to prove existence of important objects (e.g., Dvoretzky's theorem and numerous other applications of the probabilistic method to prove existence statements), and the study of the behavior of a variety of important stochastic processes in a geometric setting. In this talk we will describe a useful paradigm in metric geometry (originating from the work of Keith Ball) that allows for a probabilistic interpretation of certain geometric questions whose statement does not have any a priori connection to probability. In particular, we will address the following topics.
 Using Markov chains to show that it is possible to extend Lipschitz functions between certain metric spaces.
 Using Markov chains to prove impossibility results for Lipschitz extension problems.
 Using Markov chains to prove that for certain pairs of metric spaces X, Y, any embedding of X into Y must significantly distort distances.
 Using Markov chains to show that good embeddings do exist between certain metric spaces.
 Using Markov chains in metric Ramsey theory.
 Markov chains as an invariant for Lipschitz quotients and a tool to understand isomorphic uniform convexity.
 Markov chains as a tool to prove nonlinear spectral calculus inequalities.

A Stochastic Game of Control and StoppingWe study the existence of optimal actions in a zerosum game inf_{τ}sup_{P}E^{P}[X_{τ}] between a stopper and a controller choosing the probability measure. We define a nonlinear Snell envelope Y via the theory of sublinear expectations and show that the first hitting time inf{t:Y_{t}=X_{t}} is an optimal stopping time. The existence of a saddle point is obtained under a compactness condition. (Joint work with Jianfeng Zhang.)

Regularity Conditions in the CLT for Linear Eigenvalue Statistics of Wigner MatricesWe show that the variance of centred linear statistics of eigenvalues of GUE matrices remains bounded for large n for some classes of test functions less regular than Lipschitz functions. This observation is suggested by the limiting form of the variance (which has previously been computed explicitly), but it does not seem to appear in the literature. We combine this fact with comparison techniques following TaoVu and Erdős, Yau, et al. and a LittlewoodPaley type decomposition to extend the central limit theorem for linear eigenvalue statistics to functions in the Hölder class C^{1/2+ε} in the case of matrices of Gaussian convolution type. We also give a variance bound which implies the CLT for test functions in the Sobolev space H^{1+ε} and C^{1ε} for general Wigner matrices satisfying moment conditions. If the additional assumption of the test function being supported away from the edge of the spectrum is made, we prove the CLT for test functions of regularity Ḣ^{1/2} ∩ L^{∞} and H^{1/2+} for GUE and Johansson matrices respectively. Previous results on the CLT impose the existence and continuity of at least one classical derivative.
 All Talks
 About Heavy Tails Random Matrices
 Heat Flow, Harnack Inequalities, and Optimal Transportation
 Robust Optimality of Gaussian Noise Stability
 Geometric Applications of Markov Chains
 A Stochastic Game of Control and Stopping
 Regularity Conditions in the CLT for Linear Eigenvalue Statistics of Wigner Matrices