Variational problems in physics, economics and geometry

我建议使用最优运输来研究物理学、经济学和几何学中的问题。最优运输问题可以被漫画化为：给定买卖双方在农村的分布，将买卖双方配对，以最小化给定的运输成本。在这里，农村可以是高维或低维的，但有这么多的买家和卖家，这个问题变得难以计算，人们必须研究连续极限，其中解决方案采取最优地图的形式，其结构由成本的选择决定。这个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，这些映射何时是平滑的，如果不是，它们的不连续性是否可以被表征？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的空间的情况下。我计划在应用方面与经济学家合作，例如描述面临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前沿的形成和演变（可以在上述地图中建模为不连续性）。我还计划研究大量生物体或粒子的自组装/聚集动力学，通过一对相互作用耦合，在短距离内相互排斥，在大范围内相互吸引。这种动力学用于模拟动物的群集和群集运动;它们根据交互的选择显示各种各样的模式。我计划用最优输运的思想来发展一个非光滑的引力理论。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。它传统上将时空建模为局部光滑的，但经常预测这种光滑性必须在未来的某个有限时刻（称为奇点）破裂，例如在黑洞内部。它预测物质无法在这种崩溃中幸存下来，并且对之后发生的事情几乎没有预测。爱因斯坦的理论是用经典微分几何的语言来表达的，除了它被投射到洛伦兹空间而不是黎曼空间，这意味着空间中的所有点对不是由距离分开的;相反，有些是由时间分开的，在这种情况下，它们有一个明确的顺序-未来与过去-以及最大年龄将它们分开。通过将我最近的工作与度量测度几何的其他发展相结合，我们有可能利用概率测度的测地线的熵凸性沿着性质来解释爱因斯坦在非光滑时空中的理论。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现象的新见解。

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.