Variational problems in optimal mass transportation and intersection homology theory

摘要：本课题的目标是研究由最优质量运输和奇异变量的交点同调理论导出的几何变分问题。第一个提议的项目是建立一个模型来模拟自然界中的“树状”物体，并将其应用于最优的公共交通问题。本文提出的问题包括建立可整流变量的输运理论，寻找该模型与其他问题的联系，并将该理论推广到更一般的空间。第二个项目涉及奇异空间中的最小曲面，例如在其相交同调群下的奇异复射影变。在他的论文中，PI给出了可整流版本的关于分层次解析伪流形的交同调理论。他证明了在每一个交集同调类中都存在一个修正的质量最小器（它是一个可整流电流）。相关的质量最小值几乎是最小的电流。PI所考虑的质量最小值可能（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将继续他的研究相关的质量最小化的规律性，并可能与霍奇理论的联系。这两个项目中使用的主要方法之一是几何测量理论。“树形”路径的现象在自然界中很常见。树木、铁路、航空、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大众交通的研究领域，而且具有理论和应用的双重意义。生物学家目前正在寻找一种模型来对生物标度定律进行基本解释，生物标度定律是生物体的生物学特性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的质量运输问题与许多数学领域密切相关，尤其是偏微分方程，因此这种新的质量运输方法也有可能在经济学、生物学、图像处理和其他一些学科中得到应用。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肥皂膜的全局特性，这些特性存在于更一般的单一空间中，并可能与霍奇理论和最佳大众运输联系起来。这可能会给著名的霍奇猜想提供一些线索。

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.