Variational problems in optimal mass transportation and intersection homology theory

公共交通中的变分问题和相互作用同伦：夏庆兰摘要：拟议项目的目标是研究最优公共交通中的几何变分问题，以及奇异变种的交同调理论。第一个被提议的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规模运输问题。文中提出的问题包括建立可纠变集的传输理论，寻找该模型与其他问题的联系，以及将该理论推广到更一般的空间。第二个项目涉及居住在奇异空间中的极小曲面，如奇异复射射簇的交同调群。在他的论文中，PI给出了分层次解析伪流形上交同调理论的可导电流版本。他证明了在每个交同调类中都存在一个修正的质量极小元(这是一个可求的电流)。与之相关的质量极小化结果是几乎最小的电流。PI所考虑的质量极小可能与奇异空间的奇异轨迹相交(以受控的方式)。在这项提议中，PI将继续他对相关质量极小化的正则性的研究，并可能与Hodge理论相联系。这两个项目使用的主要方法之一是几何测度论。“树形”路径的现象在自然界中非常普遍。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种模型，以给出生物比例律的基本解释，生物比例律是生物体生物学如何随其大小变化的数学表达式。本研究提出的模型可能是理想的模型。由于Monge的质量传输问题与许多数学领域密切相关，特别是与偏微分方程有关，这种新的质量传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。肥皂膜是极小表面的物理模型。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奇异空间中的肥皂膜的全局性质，以及与Hodge理论和最优质量传输的可能联系。这可能会为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.