Variational problems in physics, economics and geometry

我建议用最优运输来研究物理、经济和几何问题。最优运输问题可以画成如下：给定买者和卖者在农村的分布，将买者和卖者配对以使给定的运输成本最小化。这里的乡村可以是高维的，也可以是低维的，但由于有太多的买家和卖家，这个问题在计算上变得难以处理，人们必须研究连续体极限，在这个极限下，解决方案采用最优地图的形式，其结构由成本选择决定。事实证明，这一理论与数学内外的许多主题都有着深刻的联系。尽管取得了很大进展，但一些基本问题仍然存在。例如，这些地图什么时候会是平滑的，如果不是，它们的不连续是否可以被表征？我建议研究这些问题，特别是在消费者和生产者生活在不同维度空间的情况下。我计划在应用方面与经济学家合作，比如描述面对信息不对称的最佳决策（激励理论），并与气象学家合作研究大气压力锋的形成和演变（可以在上述地图中建模为不连续点）。我还计划研究大量生物或粒子的自组装/聚集动力学，通过对相互作用耦合，在短距离上排斥，在大范围内吸引。这种动力学被用来模拟动物的群集和群体运动；根据交互方式的选择，它们会显示各种各样的模式。

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.