Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.
Friday, March 24, 2017
PhD Candidate: Benjamin Schachter
Supervisor: Almut Burchard
Thesis title: An Eulerian Approach to Optimal Transport with Applications to the Otto Calculus
This thesis studies the optimal transport problem with costs induced by Tonelli Lagrangians. The main result is an extension of the Otto calculus to higher order functionals, approached via the Eulerian formulation of the optimal transport problem. Open problems 15.11 and 15.12 from Villani’s Optimal Transport: Old and New are resolved. A new class of displacement convex functionals is discovered that includes, as a special case, the functionals considered by Carrillo-Slepčev. Improved and simplified proofs of the relationships between the various formulations of the optimal transport problem, first seen in Bernard-Buffoni and Fathi-Figalli, are given. Progress is made towards developing a rigourous Otto calculus via the DiPerna-Lions theory of renormalized solutions. As well, progress is made towards understanding general Lagrangian analogues of various Riemannian structures.
A copy of the thesis can be found here: DraftThesisSchachter