FRG: Collaborative Research: Applications of Transportation Theory to Nonlinear Dynamics

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Proposal DMS-0354729PIs: W.Gangbo (Georgia Tech), L.Caffarelli (U Texas), L.C.Evans(U California Berkeley), M. Feldman (U Wisconsin), R.McCann (U. Toronto)Title: Applications of Transportation Theory to Nonlinear DynamicsABSTRACTThis project focuses on the analysis of a collection of variationaloptimization and dynamical evolution problems centered around the themeof optimal transportation --- which enters the dynamical settingwhenever 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 totaltransportation costs. Here the cost per ton of ore transported from themine 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 andfactories are distributed continuously throughout Euclidean space ora curved landscape with obstacles --- and the cost is related to thedistance on this landscape, then the problem has a rich structure anddeep connections to geometry and non-linear PDE which have only begun to be explored. Incarnations of this problem embed in current models forsurprisingly diverse phenomena. Along with basic questions concerning the structure and qualitative features of optimal mappings, the proposedresearch 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 aninternational 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 participantsaccess 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.