PIMS-UBC Distinguished Colloquium: Young-Heon Kim

This talk brings together ideas from optimization, probability, and partial differential equations.

Classical optimal transport asks how to move one prescribed distribution of mass to another at minimal cost. Beginning with the concrete problem of transporting mass efficiently, it has developed into a unifying framework connecting many areas of mathematics, including partial differential equations, probability, geometry, and mathematical physics, with more recent applications in data science and machine learning.

I will discuss a class of stochastic optimal transport problems in which mass moves along random trajectories and the transport is governed by an optimal stopping rule. The associated stopping problems naturally give rise to free boundaries in space and time.

I will then turn to free-target problems, in which the terminal distribution is not prescribed in advance but is selected by the optimization, subject to a density constraint. A central mechanism is saturation of this constraint, which identifies the resulting stochastic transport evolution with a Stefan-type free boundary problem modelling phase transitions between water and ice.

The main example is the supercooled Stefan problem, in which supercooled water freezes along an unstable moving interface. Historically, most results have concerned one-dimensional settings, while higher-dimensional existence theory has largely been confined to special situations such as radial symmetry. I will explain how the free-target viewpoint provides a variational construction of global-in-time weak solutions in general dimensions for a broad class of initial data, together with a natural stochastic principle for selecting a distinguished solution.