Thursday, May 19, 2011, 1:00 p.m.,
in Room BA 6183, 40 St. George Street

PhD Candidate:  Brendan Pass

PhD Advisor:  Robert McCann

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.


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