Wilfrid Gangbo, UCLA

Abstract: This is an introductory talk on optimal transport theory, which in the past two decades, has emerged as a fertile field of inquiry and a powerful tool for applications to problems within and beyond mathematics. The physical interpretation of the basic issue is this. We ask how we can optimally rearrange a given pile of soil or rubble (the "deblais") with mass distribution u0, into an excavation or fill (the "remblais"), with mass distribution u1? Here, optimality is measured against the cost function c(x, y) = ||x-y||p: A geometric characterization of the solutions allows to describe geodesics of minimal lengths on the set of probability measures and to derive sharp functional inequalities.