# Mathematics Colloquia and Seminars

Optimal Transport is the study of moving some quantity, be it physical or abstract, from one distribution to another, and finding a strategy for achieving the desired transport that minimizes the total cost according to some cost function. While the classical analysis in Optimal Transportation involves problems with binary cost functions $c(x,y)$ that look at only where ‘mass’ starts and ends, another important class of problems are path-dependent. Minimizing solutions of the path-dependent $M_\alpha$ transport cost often involve ramified paths, and their study allows for analysis of branched transport systems with useful application to a wide range of phenomena. In this talk we will examine the fundamentals of this Ramified Optimal Transportation theory and briefly survey some of its applications, then we will finally consider the “Landscape” functions associated to optimal paths (the marginal cost of transport, or first variation of the $M_\alpha$ cost) and look at directions for possibly extending Landscape function theory beyond its current development. No prior exposure to optimal transport theory is assumed.