This project focuses on the analysis of a collection of variational optimization and dynamical evolution problems centered around the theme of optimal transportation --- which enters the dynamical setting whenever the evolution conserves a scalar locally. The central problem can be caricatured as follows:Given a distribution of iron mines throughout the countryside, and a distribution of factories which require iron ore, decide which mines should supply ore to each factory in order to minimize the total transportation costs. Here the cost per ton of ore transported from the mine at x to factory at y is specified by a function c(x,y) --- so the problem can be formulated as a linear program. However, when the mines and factories are distributed continuously throughout Euclidean space or a curved landscape with obstacles --- and the cost is related to the distance on this landscape, then the problem has a rich structure and deep connections to geometry and non-linear PDE which have only begun to be explored. Incarnations of this problem embed in current models for surprisingly diverse phenomena. Along with basic questions concerning the structure and qualitative features of optimal mappings, the proposed research addresses models for front formation in the atmosphere, dissipative equilibration in kinetic theory, fluid flow, and granular materials and geometric and dynamical inequalities.
After half a century of mathematical neglect, the past decade witnessed a revival of interest in optimal transportation, and watched as it blossomed into a fertile field of investigation as well as a vibrant tool for exploring diverse applications within and beyond mathematics. The transformation occurred partly because long-standing issues could finally be resolved, but also because unexpected connections were discovered which linked these questions to problems in physics, geometry, computer vision, partial differential equations, earth science and economics. The time is ripe for a collaborative effort on an international scale to explore existing connections and unearth new ones, while simultaneously developing the basic theory of optimal maps and introducing students and colleagues to the challenges and promise of the field --- thus for the formation of a focused research group with these goals. The core of our plan is to arrange sustained interactions between and around members of the group, who in addition to collaborating scientifically, will work together over the next several years to create the research environment and manpower necessary for transportation research to flourish. To achieve this goal, we plan to organize two workshops on different aspects of the subject. Furthermore, we plan to share the responsibilities of training graduate students and postdoctoral fellows, by using funds from the grant to support young researchers. This unique arrangement will give participants access to an unusually broad assortment of perspectives and expertise. Moreover, we believe a three-year nurturing window for young researchers to learn the subject and become involved --- if continued now --- will ultimately further advance progress in the field by more than a decade.
