## Departmental PhD Thesis Exam – Brendan Pass

Thursday, April 7, 2011, 12:10 p.m.,
in Room 210, Fields Institute, 222 College St.

PhD Candidate:  Brendan Pass

PhD Thesis Title: Structural results on optimal transportation plans
(http://www.math.utoronto.ca/bpass/ut-thesis.pdf)

PhD Thesis Abstract:

In this thesis we prove several results on the structure of solutions to
optimal transportation problems.

The second chapter represents joint work with Robert McCann and Micah Warren;
the main result is that, under a non-degeneracy condition on the cost function, the
optimal is concentrated on a n-dimensional Lipschitz submanifold of the product space.
As a consequence, we provide a simple, new proof that the optimal map satisfies a
Jacobian equation almost everywhere. In the third chapter, we prove an analogous result
for the multi-marginal optimal transportation problem; in this context, the dimension
of the support of the solution depends on the signatures of a family of semi-Riemannian
metrics on the product space. In the fourth chapter, we identify sufficient conditions
under which the solution to the multi-marginal problem is concentrated on the graph of
a function over one of the marginals. In the fifth chapter, we investigate the regularity
of the optimal map when the dimensions of the two spaces fail to coincide. We prove that
a regularity theory can be developed only for very special cost functions, in which case a
quotient construction can be used to reduce the problem to an optimal transport problem
between spaces of equal dimension. The final chapter applies the results of chapter 5 to
the principal-agent problem in mathematical economics when the space of types and the
space of products differ. When the dimension of the space of types exceeds the dimension
of the space of products, we show if the problem can be formulated as a maximization
over a convex set, a quotient procedure can reduce the problem to one where the two
dimensions coincide. Analogous conditions are investigated when the dimension of the
space of products exceeds that of the space of types.