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Explore a 51-minute lecture by Dmitry Vorotnikov from the University of Coimbra on matrix-valued problems similar to optimal transport and their applications to partial differential equations. Recorded at IPAM's Dynamics of Density Operators Workshop, the talk examines how the Benamou-Brenier representation of optimal transport problems has a dual formulation involving the quadratic Hamilton-Jacobi equation. Learn how reversing this perspective allows recovery of optimal transport problems from Hamilton-Jacobi equations through duality, and how this procedure applies to various PDEs. Discover how, except in certain degenerate cases including classical optimal transport, the transported probability measure is automatically replaced by a non-negative-definite matrix-valued measure. The presentation covers Brenier's 2018 demonstration of using this approach to construct solutions to Cauchy problems for PDEs, recent developments in the field, and connections to Dafermos' selection principle.
