Rich Jordan: Abstract

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The mass transportation, or "dirt mover's" problem goes back to Gaspard
Monge (1781) and can be roughly stated as: How can we move a pile of
dirt occupying a region X to a region Y in such a way as to minimize
the transportation cost? Kantorovich received a Nobel Prize in
economics for work related to this problem.

The Fokker-Planck, or forward Kolmogorov equation, describes the
dynamics of the probability density of a particle undergoing diffusion
in a potential.

How are these two problems related?

The answer lies in a surprising variational formulation of the
Fokker-Planck equation, which allows us to interpret the dynamics as a
gradient flow, or steepest descent, of the system free energy in the
Wasserstein metric.  The latter induces a natural topology on the space
of probability measures, and also provides the connection between
diffusion and optimal transport.

In the last 10 years or so, there has been an explosion of research on
gradient flow forulations of linear and nonlinear diffusion equations
via the Wasserstein metric framework.  In this talk, we'll see where
and when it got started and where it has gone/is going.  Most recently,
for example, researchers have been using this approach in connection
with Ricci curvature.