What Is Measure Transportation and Why Should We Care?

Speaker: Markos Katsoulakis (Umass)

Abstract:

Measure transportation studies how to move one probability distribution into another in an optimal or geometrically meaningful way. Originating in the work of Monge and Kantorovich, it has become a central mathematical framework connecting analysis, geometry, probability, and partial differential equations.

In this talk, I will introduce the basic ideas behind transporting probability measures and explain how Wasserstein distances provide a geometric notion of distance between distributions. I will also briefly discuss information divergences, such as relative entropy, as an alternative way to compare and redistribute probability mass, and how these perspectives jointly lead naturally to gradient flows in probability space and variational time discretizations.

Finally, I will discuss modern regularized and proximal formulations that make these methods stable and computationally tractable in high dimensions. These ideas now play an important role in machine learning and in applications across science, engineering, and medicine, including reconstructing cellular dynamics from single-cell RNA sequencing data.

The goal of the talk is to provide intuition, mathematical structure, and to illustrate how rigorous mathematics in measure transportation fundamentally shapes modern machine learning and data-driven predictive modeling across science, engineering, and medicine.