The optimal transport problem, in its Benamou-Brenier formulation, admits a dual representation involving the quadratic Hamilton-Jacobi equation. Viewed in reverse, this perspective allows one to recover the
optimal transport problem from the Hamilton-Jacobi equation via duality. A similar procedure can be applied to many other PDEs and even to algebraic equations. Remarkably, except in certain degenerate cases (including, of course, the classical optimal transport), the transported probability measure is naturally replaced by a positive-semidefinite-matrix-valued measure. In 2018, Brenier demonstrated how this duality approach can be employed to construct solutions to Cauchy problems for PDEs. We will discuss some recent developments and applications.