Variational problems in physics, economics and geometry

点击翻译按钮获取中文摘要

I propose to study problems in Physics, Economics and Geometry using Optimal Transport. The optimal transport problem can be caricatured as follows: given distributions of buyers and sellers over the countryside, pair buyers with sellers so as to minimize a given transportation cost. Here the countryside can be high or low dimensional, but there are so many buyers and sellers that the problem becomes computationally intractable, and one has to study the continuum limit, where the solution takes the form of an optimal map whose structure is determined by the choice of cost. This theory turns out to have deep connections to many topics both within and outside mathematics. Despite much progress, basic questions still remain. For example, when will these maps be smooth, and if not, can their discontinuities be characterized? I propose to study such questions, especially in the case where the consumers and producers live in spaces of different dimension. I plan to collaborate with economists on applications, such as characterizing optimal decisions facing informational asymmetry (the theory of incentives), and with meteorologists on the formation and evolution of atmospheric pressure fronts (which can be modelled as discontinuities in the aforementioned maps). I also plan to study the self-assembly/aggregation dynamics of a large number of organisms or particles, coupled by a pair interaction which repels at short distances and attracts at large. Such dynamics are used to model the swarming and flocking of animals and crowd motion; they display a wide variety of patterns depending on choice of interaction. I plan to use optimal transport ideas to develop a nonsmooth theory of gravity. The Einstein field equation is one of the most fundamental equations of physics. It relates the bending of spacetime to the energy and momentum of matter. It traditionally models spacetime as being locally smooth, yet often predicts that this smoothness must breakdown at some finite moment in the future - called a singularity - for example in the interior of a blackhole. It predicts that physical matter cannot survive this breakdown, and makes few predictions for what happens afterwards. Einstein's theory is expressed in the language of classical differential geometry, except that it is cast into Lorentzian rather than Riemannian spaces, meaning not all pairs of points in the space are separated by distance; instead some are separated by time, in which case they have a definite ordering - future versus past - and maximum age separating them. By combining my recent work with other developments in metric measure geometry, it is possible to use entropic convexity properties along geodesics of probability measures to give a sense to Einstein's theory in nonsmooth spacetimes. I plan to develop this point of view and explore its consequences, which has the potential to lead to new insight into phenomena ranging from black hole dynamics to the structure of the universe.