Some geometric variational problems on ramified transportation and intersection homology

分支交通与交叉同调的一些几何变分问题

• 批准号: 0710714
• 负责人: Qinglan Xia
• 金额: $ 9.97万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0710714&HistoricalAwards=false
• 关键词: Some; geometric; variational; problems; ramified

This project concerns geometric variational problems derived from optimal transportation as well as from the intersection homology theory of singular varieties. The first part of the project studies optimal ramified (i.e. branching) transportation, which was modeled after many branching structures in nature such as trees, river channel networks etc. A central concept there is the notion of optimal transport path, which mathematically plays the role of a "geodesic" between two probability measures. Graphically, it has a "tree-shaped" branching structure similar to fractals. Some of the problems to be addressed within the project are the following: (1) Optimal ramified transportation in metric measure spaces; (2) Optimal ramified transportation of higher dimensional geometric objects (3) Flow of surfaces driven by curvature and transportation (including both ramified type transportation and Monge-Kantorovich type transportation), which are modeled after the growth of tree leaves, flowers and mud-cracking; (4) Dimensional distance between measures and sets. The second part of the project studies properties of "minimal surfaces" lived in singular varieties, using geometric measure theory. The PI proved that the intersection homology theory of MacPherson and Gorsky on singular varieties can be reformulated in terms of rectifiable currents, and also studied existence as well as partial regularities of minimizers under suitable modified masses within each intersection homology group. The PI aims at further investigations of these minimizers using analytic tools developed in geometric measure theory.One of the main purposes of mathematics is to explore the beauty and mechanisms of many naturally generated shapes such as soap films, trees, leaves, and mud cracking. Many times, optimization process plays a very important role in the formation of these shapes. For instance, soap films come from minimizing surface area, while trees adopt branching transport systems to minimize transportation cost. This project aims at studying geometric shapes whose formation is driven by some optimization process. The first part of this project studies "tree-type" branching systems, which are commonly found in living and non-living systems such as trees, railways, river channel networks, lightning, the circulatory system, and neural networks. Nature has selected these branching transport systems partially because they are comparably cost efficient. As a result, to study mechanisms behind the formation of these branching systems, one may simply start with the study of optimal transport systems. In the last few years, the PI and many others have developed geometric variational methods for studying optimal ramified transportation, and also found its application in modeling the formation of tree leaves. In this project, the principal investigator is interested in developing the existing theory in a more general setting, and then applies it to transport specific geometric objects. Also, by considering geometric flow of surfaces driven by curvature and transportation, one may use them to model the formation of leaves, flowers, and some other fractal type objects. On the other hand, soap films are physical model for minimal surfaces, which play an important role as a tool in the study of topology, geometry and physics. The second part of the project aims at studying general properties of "soap films" lived in singular varieties, which are very nice but usually contains some singularities, within their intersection homology groups. We will mainly use geometric measure theory as our analytic tools.

本课题涉及最优运输中的几何变分问题，以及奇异变种的交同调理论。该项目的第一部分研究最优分叉(即分支)运输，它是根据自然界中的许多分支结构(如树木、河道网络等)模拟的。最优运输路径的一个核心概念是最优运输路径的概念，它在数学上扮演着两个概率度量之间的“测地线”的角色。从图形上看，它有一个类似于分形图的“树形”分支结构。该项目要解决的一些问题如下：(1)度量空间中的最优分枝运输；(2)高维几何对象的最优分枝运输；(3)曲率和运输(包括分支型运输和Monge-Kantorovich型运输)驱动的表面流，它们是根据树叶、花朵和泥浆裂缝的生长来模拟的；(4)度量和集合之间的维度距离。该项目的第二部分利用几何测度论研究了奇异簇中的“极小曲面”的性质。PI证明了MacPherson和Gorsky关于奇异簇的交同调理论可以用可导电流来表示，并在每个交同调群内研究了在适当的修正质量下极小元的存在性和部分正则性。PI旨在利用几何测量理论中发展起来的分析工具进一步研究这些极小化。数学的主要目的之一是探索许多自然生成的形状的美和机理，如肥皂膜、树、树叶和泥浆裂缝。很多时候，优化过程在这些形状的形成中起着非常重要的作用。例如，肥皂膜来自于最大限度地减少表面积，而树木采用分枝运输系统来最大限度地降低运输成本。这个项目的目的是研究几何形状的形成，这些几何形状是由某种优化过程驱动的。该项目的第一部分研究“树型”分支系统，这种系统通常存在于生物和非生物系统中，如树木、铁路、河网、闪电、循环系统和神经网络。大自然之所以选择这些分支运输系统，部分原因是它们具有相对的成本效益。因此，要研究这些分支系统形成背后的机制，可以简单地从最优运输系统的研究开始。在过去的几年里，PI和其他许多人发展了几何变分方法来研究最优分枝运输，并将其应用于树叶形成的模拟。在这个项目中，首席研究人员感兴趣的是在更一般的环境中发展现有的理论，然后将其应用于运输特定的几何对象。此外，通过考虑由曲率和运输驱动的曲面的几何流动，可以使用它们来模拟树叶、花朵和其他一些分形型对象的形成。另一方面，肥皂膜是极小曲面的物理模型，在拓扑学、几何学和物理学的研究中起着重要的作用。该项目的第二部分旨在研究生活在奇异变种中的“肥皂膜”的一般性质，这种变种非常好，但通常包含一些奇点，在它们的相交同调群中。我们将主要使用几何测度论作为分析工具。