Variational problems in optimal mass transportation and intersection homology theory

质量输运和相交同调中的变分问题PI：夏青兰摘要：本课题的目标是研究由最优质量输运和奇异簇的相交同调理论导出的几何变分问题.第一个项目是建立一个模型来模拟自然界中的“树形”物体，并将其应用于最优大众运输问题。这里提出的问题包括建立一个运输理论的可整流varifolds，找到这个模型与其他问题的连接，并将这个理论扩展到更一般的空间。第二个项目关注的极小曲面，生活在奇异空间，如奇异复射影簇下的交叉同源群。在他的论文中，PI给出了分层次解析伪流形上的交同调理论的可求正流版本。他表明，存在一个修改的质量极小（这是一个整流电流）在每一个交叉同源类。与之相关的质量最小化者几乎是最小的电流。PI所考虑的质量最小化器可以（以受控的方式）与奇异空间的奇异轨迹相交。在这个提议中，PI将继续研究相关质量最小化的规律性，以及与霍奇理论的可能联系。在这两个项目中使用的主要方法之一是几何测量理论。 "树形“路径现象在自然界中是非常普遍的。树木、铁路、航空公司、闪电、循环系统和神经网络都是常见的例子。最优“树形”路径的研究不仅拓展了公共交通的研究领域，而且具有重要的理论意义和应用价值。生物学家目前正在寻找一个模型，以给出生物尺度定律的基本解释，生物尺度定律是生物体生物学如何随其大小而变化的数学表达式。本研究提出的模型可能是理想的模型。由于Monge的质量输运问题与数学的许多领域，特别是与偏微分方程密切相关，因此这种新的质量输运方法也可能在经济学、生物学、图像处理等学科中得到应用。肥皂膜是最小表面的物理模型。这些曲面在拓扑学、几何学和物理学的研究中起着重要的作用。第二个项目的研究涉及生活在更一般的奇异空间和可能的联系霍奇理论和最优质量运输的肥皂膜的全局属性。这可能为著名的Hodge猜想提供一些线索。

VARIATIONAL PROBLEMS IN MASS TRANSPORTATION AND INTERSECTIONHOMOLOGYPI: QINGLAN XIA ABSTRACT: The goal of the proposed projects is to study geometric variational problems derived from optimal mass transportation as well as from the intersection homology theory of singular varieties. The first proposed project is building a model to simulate ``tree shaped'' objects in nature, and apply it to optimal mass transportation problems. The problems proposed here include building a transport theory of rectifiable varifolds, finding connections of this model with other problems, and extending this theory to more general spaces. The second project concerns minimal surfaces that live in singular spaces such as singular complex projective varieties under their intersection homology groups. In his thesis, the PI gave a rectifiable currents' version of intersection homology theory on stratified subanalytic pseudomanifolds. He showed that there exists a modified mass minimizer (which is a rectifiable current) in every intersection homology class. The associated mass minimizers turn out to be almost minimal currents. The mass minimizers the PI considered may intersect (in a controlled fashion) the singular locus of the singular space. In this proposal, the PI will continue his investigation on regularity properties of the associated mass minimizers, and possible link with Hodge theory. One of the main methods used in both projects is geometric measure theory. The phenomenon of ``tree shaped'' path is very common in nature. Trees, railways, airlines, lightning, the circulatory system, and neural networks are common examples. The research of optimal ``tree shaped'' path not only expands the research area of mass transportation, but also of both theoretical and applied interest. Biologists are currently looking for a model to give a fundamental explanation for biological scaling laws which are mathematical expressions of how organisms' biology varies with their size. The model proposed in this research might be the desired one. As Monge's mass transport problem is strongly linked with many areas of mathematics, especially with partial differential equations, it is also possible that this new approach of mass transportation will find its applications in economics, biology, image processing and some other subjects. Soap films are physical model for minimal surfaces. These surfaces play an important role as a tool in the study of topology, geometry and physics. The research of the second project concerns global properties of soap films that live in more general singular spaces and possible links to Hodge theory and optimal mass transportation. It might provides some hint to the famous Hodge conjecture.