Geometry on the Set of Probability Measures

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Geometry on the Set of Probability Measures.Abstract of Proposed ResearchWilfrid Gangbo This project is to analyze certain variational problems connected with partial differential equations and applications. A primary topic is the Monge-Kantorovich mass transportation problem and its applications. Also the theory of infinite dimensional Hamiltonian systems on certain spaces of probability measures including examples such as Vlasov-type equations and the semi-geostrophic equation. Another class of problems are the systems of nonlinear parabolic equations that arise in elasticity theory and the design of frames and structures which are optimal in certain senses. The problems to be studied in this project have applications in many different areas including economics, physics, geosciences and engineering. The aim is always to identify the optimal way for a process to occur; so the results may be very important and have wide applicability.