Variational problems in physics, economics and geometry

我建议用最优运输来研究物理学、经济学和几何学中的问题。最优运输问题可以刻画为：给定农村地区买家和卖家的分布，将买家和卖家配对，使给定的运输成本最小。这里的乡村可以是高维的，也可以是低维的，但有太多的买家和卖家，这个问题在计算上变得难以处理，人们必须研究连续统的极限，其中解的形式是最优地图的形式，其结构由成本的选择决定。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，但基本问题仍然存在。例如，这些地图何时会是平滑的，如果不是，它们的不连续性能否被刻画出来？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的空间的情况下。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优决策的特征(激励理论)，以及与气象学家合作研究大气压力锋的形成和演变(可以在前述地图中建模为不连续)。我还计划研究大量有机体或粒子的自组装/聚集动力学，加上一对相互作用，这种相互作用在短距离内相互排斥，并在总体上相互吸引。这种动力学被用来模拟动物的成群和成群以及人群的运动；它们根据互动的选择显示出各种各样的模式。我计划用最优运输的想法来发展一种非光滑的重力理论。爱因斯坦场方程是物理学中最基本的方程之一。它将时空的弯曲与物质的能量和动量联系起来。它传统上把时空建模为局部光滑的，但经常预测这种光滑在未来的某个有限时刻--称为奇点--必然会破裂，例如在黑洞的内部。它预测物理物质无法在这种分裂中幸存下来，而对随后发生的事情几乎没有做出预测。爱因斯坦的理论是用经典微分几何的语言表达的，只是它被投射到洛伦兹空间而不是黎曼空间，这意味着空间中并不是所有的点对都被距离隔开；相反，有些点是被时间隔开的，在这种情况下，它们有一个明确的顺序--未来和过去--以及分隔它们的最大年龄。通过将我最近的工作与度规几何的其他发展相结合，有可能使用概率度量的测地线上的熵凸性性质来赋予爱因斯坦在非光滑时空中的理论意义。我计划发展这一观点并探索其后果，这有可能导致对从黑洞动力学到宇宙结构等各种现象的新洞察。

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.