Georgia Institute of Technology College of Sciences

Burgers turbulence is the study of Burgers equation with random initial data or forcing. While having its origins in hydrodynamics, this model has remarkable connections to a variety of seemingly unrelated problems in statistics, kinetic theory, random matrices, and integrable systems. In this talk I will survey these connections and discuss the crucial role that exact solutions have played in the development of the theory.

Rank reduction as an effective technique for dimension reduction is widely used in statistical modeling and machine learning. Modern statistical applications entail high dimensional data analysis where there may exist a large number of nuisance variables. But the plain rank reduction cannot discern relevant or important variables. The talk discusses joint variable and rank selection for predictive learning. We propose to apply sparsity and reduced rank techniques to attain simultaneous feature selection and feature extraction in a vector regression setup. A class of estimators is introduced based on novel penalties that impose both row and rank restrictions on the coefficient matrix. Selectable principle component analysis is proposed and studied from a self-regression standpoint which gives an extension to the sparse principle component analysis. We show that these estimators adapt to the unknown matrix sparsity and have fast rates of convergence in comparison with LASSO and reduced rank regression. Efficient computational algorithms are developed and applied to real world applications.

Ricci flow is a sort of (nonlinear) heat problem under which the metric on a given manifold is evolving. There is a deep connection between probability and heat equation. We try to setup a probabilistic approach in the framework of a stochastic target problem. A major result in the Ricci flow is that the normalized flow (the one in which the area is preserved) exists for all positive times and it converges to a metric of constant curvature. We reprove this convergence result in the case of surfaces of non-positive Euler characteristic using coupling ideas from probability. At certain point we need to estimate the second derivative of the Ricci flow and for that we introduce a coupling of three particles. This is joint work with Rob Neel.

The Burgers equation is a basic hydrodynamic model describing the evolution of the velocity field of sticky dust particles. When supplied with random forcing it turns into an infinite-dimensional random dynamical system that has been studied since late 1990's. The variational approach to Burgers equation allows to study the system by analyzing optimal paths in the random landscape generated by the random force potential. Therefore, this is essentially a random media problem. For a long time only compact cases of Burgers dynamics on the circle or a torus were understood well. In this talk I discuss the Burgers dynamics on the entire real line with no compactness or periodicity assumption. The main result is the description of the ergodic components and One Force One Solution principle on each component. Joint work with Eric Cator and Kostya Khanin.