Collaborative Research: Optimal Transportation: Its Geometry and Applications

摘要：最优运输：它的几何及其应用这个项目集中分析了一系列变分优化和动态进化问题，这些问题围绕着最优运输的主题-当进化保持局部标量时，进入动态设置。核心问题可以概括为：鉴于铁矿在农村的分布，以及需要铁矿石的工厂的分布，决定哪些矿山应该向每个工厂供应铁矿石，以将总运输成本降至最低。这里，从x处的矿山运输到y处的工厂的每吨矿石的成本由一个函数c(x，y)指定-所以这个问题可以用一个线性规划来表示。然而，当矿山和工厂连续分布在欧几里德空间或具有障碍物的弯曲景观中-成本与该景观上的距离有关时，问题就具有丰富的结构和与几何和非线性PDE的深刻联系，而这些才刚刚开始探索。这个问题的化身嵌入到了当前的模型中，产生了令人惊讶的各种现象。除了与最优映射的结构和质量特征有关的基本问题外，拟议的研究还涉及大气中锋面形成的模型、动力学理论中的耗散平衡、流体流动、弹性晶体和颗粒材料、几何和动力学不平等，以及在委托-代理框架中形成的微观经济决策问题，这些问题涉及在信息不对称的情况下设计价格制度、税收结构或合同。在经历了半个世纪的数学忽视之后，过去十年见证了对最优交通的兴趣的复苏，看着它发展成为一个肥沃的研究领域，以及一个探索数学内外各种应用的充满活力的工具。这种转变之所以发生，部分是因为长期存在的问题终于可以得到解决，但也因为发现了意想不到的联系，将这些问题与物理、几何、计算机视觉、偏微分方程式、地球科学和经济学的问题联系在一起。现在是时候在国际范围内开展合作，探索现有的联系并挖掘新的联系，同时发展最优地图的基本理论，并向学生和同事介绍该领域的挑战和前景-从而形成一个具有这些目标的重点研究小组。我们计划的核心是安排小组成员之间和周围的持续互动，他们除了进行科学合作外，还将在未来几年共同努力，为交通研究的蓬勃发展创造必要的研究环境和人力。为了实现这一目标，我们计划在我们的几个国内机构组织一系列为期三个学期的长时间的重点课程和两个关于这门学科的不同方面的研讨会。此外，我们计划分担培训研究生和博士后研究员的责任，使用补助金中的资金支持年轻研究人员，同时允许他们在本国机构和重点学习的学期之间分配时间。这一独特的安排将使与会者有机会接触到异常广泛的观点和专业知识。此外，我们相信，为年轻研究人员提供三年的培养窗口，让他们学习这一主题并参与其中--如果现在建立--最终将使该领域的进展提前十年以上。

ABSTRACT:Optimal Transportation: Its Geometry and ApplicationsThis project focuses on the analysis of a collection of variational optimization and dynamical evolution problems centered around the theme of optimal transportation --- which enters the dynamical setting whenever the evolution conserves a scalar locally. The central problem can be sketched 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 total transportation costs. Here the cost per ton of ore transported from the mine at x to factory at y is specified by a function c(x,y) --- so the problemcan be formulated as a linear program. However, when the mines and factories are distributed continuously throughout Euclidean space or a curved landscape with obstacles --- and the cost is related to the distance on this landscape, then the problem has a rich structure and deep connections to geometry and non-linear PDE which have only begun to be explored. Incarnations of this problem embed in current models for surprisingly diverse phenomena. Along with basic questions concerning the structure and qualitative features of optimal mappings, the proposed research addresses models for front formation in the atmosphere, dissipative equilibration in kinetic theory, fluid flow, elastic crystals, and granular materials, geometric and dynamical inequalities, and microeconomic decision problems formulated in the principal-agent framework which involve designing price systems, tax structures, or contracts in the face of informational asymmetry.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 an international scale to explore existing connections and unearth new ones, while simultaneously developing the basic theory of optimalmaps 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 a series of three semester long periods of emphasis and two workshops on different aspects of the subject in several of our home institutions. Furthermore, we plan to share the responsibilities of training graduate students and postdoctoral fellows, by using funds from the grant to support young researchers while allowing them to divide their time between their home institutions and the semesters of emphasis. This unique arrangement will give participants access 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 established now - will ultimately advance progress in the field by more than a decade.