Collaborative Research: Optimal Transportation: Its Geometry and Applications

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ABSTRACT:Optimal Transportation: Its Geometry and ApplicationsThis 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 sketched 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 problemcan 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, elastic crystals, and granular materials, geometric and dynamical inequalities, and microeconomic decision problems formulated in the principal-agent framework which involve designing price systems, tax structures, or contracts in the face of informational asymmetry.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 optimalmaps 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 a series of three semester long periods of emphasis and two workshops on different aspects of the subject in several of our home institutions. Furthermore, we plan to share the responsibilities of training graduate students and postdoctoral fellows, by using funds from the grant to support young researchers while allowing them to divide their time between their home institutions and the semesters of emphasis. 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 established now - will ultimately advance progress in the field by more than a decade.