Calculus of variations and optimal mass transportation: theory and applications

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This proposal is focused on two primary directions, partial differential equations with an emphasis on the calculus of variations, and the theory and applications of optimal mass transportation. Partial differential equations (PDEs) are of widespread interest because of their connection with phenomena in the physical world. Scientists and mathematicians have become actively involved in the study of countless problems modeled by PDEs.  I am mainly interested in the study of non-linear partial differential equations arising from natural sciences and differential geometry. In addition to developing new techniques, I apply tools from functional and convex analysis to study the qualitative properties of deterministic and stochastic partial differential equations. I also study the theory and application of optimal mass transportation. Optimal mass transport is a subject with a long history, with incredibly rich theory and applications, that defines robust metrics between probability distributions. The computation of optimal displacements between distributions through an associated transport plan makes this theory mainstream in several applicable fields including image processing, cancer detection, and statistical machine learning. It is a dynamic and growing area that has led to surprising new discoveries as well as novel interpretations of classical results. The purpose of this proposed research program is two-fold. First, to combine techniques from the calculus of variations and variational inequalities with newly established variational principles to study and analyze the existence, and qualitative properties of solutions for a large class of stochastic and deterministic partial differential equations arising from the natural sciences. Second, to characterize and scrutinize solutions of standard and martingale multi-marginal optimal mass transport problems given their abilities to provide accurate models for signal intensities and other data distributions. In addition to developing the mathematical theory of optimal transportation, I will also work on its various real-life applications. I expect the outcomes of this research to have significant impacts in the fields of partial differential equations, geometric measure theory, calculus of variations, stochastic PDEs, and applications of optimal mass transportation in other fields.