May 22: Vivek Borkar, Indian Institute of Technology

Title: Averaging in two time scale stochastic systems

Abstract: I shall briefly recall some earlier joint work with Vladimir Gaitsgory on averaging in two time scale controlled diffusions and with reference to it, describe some recent work on Stackelberg stochastic differential games.

April 17: Jose Blanchet (Stanford University)

Title: Distributionally Robust Optimization via Optimal Transport and Its Applications

Abstract: Optimal mass transportation is a powerful tool in the arsenal of many quantitative disciplines, with well-documented applications spanning a wide range of areas, including, operations research, economics and image analysis. In this talk, we focus on data-driven distributionally robust optimization, that is, a class of perfect-information games in which an optimizer selects an action and adversary chooses a model within a region around a baseline distribution, which we often take to be an empirical measure. We show how many machine learning algorithms can be retrieved as special cases of this type of formulation. We establish connections to regularized portfolio optimization strategies that are common in practice. These connections provide a rich intuition which allows interpreting various regularization parameters which are typically chosen in practice via cross-validation, but owing to this intuition, we are able to define a reasonable optimization criterion for choosing regularization parameters via pivotal statistics, thereby avoiding time-consuming cross-validation.

(This talk is based on joint work with Yang Kang, Karthyek Murthy and Fan Zhang).

Info: Here are two papers which are the basis for the talk: https://arxiv.org/abs/1604.01446, https://arxiv.org/abs/1610.05627.

April 12: Susan Friedlander (Univ. of Southern California)

Title: Asymptotics for magnetostrophic turbulence in the Earth's fluid core

Abstract: We consider the 3 dimensional magnetohydrodynamics (MHD) equations in the presence of stochastic forcing as a model for magnetostrophic turbulence. For scales relevant to the Earth’s fluid core this MHD system is very rich in small parameters. We discuss results concerning the asymptotics of the stochastically forced PDEs in the limit of vanishing parameters. In particular, we establish that the system sustains ergodic statistically steady states thus providing a rigorous foundation for magnetostrophic turbulence.

This is joint work with Juraj Foldes, Nathan Glatt-Holtz and Geordie Richards.

February 8: Wilfrid Gangbo (UCLA)

Title: A partial Laplacian as an infinitesimal generator on the Wasserstein space

Abstract: We study stochastic processes on the Wasserstein space, together with their infinitesimal generators. One of these processes plays a central role in our work. Its infinitesimal generator defines a partial Laplacian on the space of Borel probability measures, and we use it to define heat flow on the Wasserstein space. We verify a distinctive smoothing effect of this flow for a particular class of initial conditions. To this end, we will develop a theory of Fourier analysis and conic surfaces in metric spaces. We note that the use of the infinitesimal generators has been instrumental in proving various theorems for Mean Field Games, and we anticipate they will play a key role in future studies of viscosity solutions of PDEs in the Wasserstein space.