Some geometric variational problems on ramified transportation and intersection homology

这个项目涉及几何变分问题，来自最优运输以及从交叉同源理论的奇异品种。该项目的第一部分研究最佳分歧（即分支）运输，这是仿照许多分支结构的性质，如树木，河道网络等一个中心概念有最佳运输路径的概念，这在数学上起着“测地线”的作用之间的两个概率措施。从图形上看，它具有类似于分形的“树形”分支结构。本课题的主要研究内容如下：（1）度量测度空间中的最优分歧输运;（2）高维几何对象的最优分歧输运;（3）由曲率和输运驱动的曲面流（包括分支式运输和蒙格-康托洛维奇式运输），以树叶、花朵和泥土的生长为模型;（4）测度与集合之间的量纲距离。 该项目的第二部分使用几何测度理论研究奇异簇中的“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求积流的形式重新表述，并研究了在每个交同调群内适当修改的质量下极小元的存在性和部分可求性。PI的目的是使用几何测量理论中开发的分析工具来进一步研究这些极小化器。数学的主要目的之一是探索许多自然生成的形状（如肥皂膜、树木、树叶和泥裂）的美丽和机制。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，肥皂膜来自最小化表面积，而树木采用分支运输系统以最小化运输成本。这个项目的目的是研究几何形状的形成是由一些优化过程驱动。该项目的第一部分研究“树型”分支系统，这是常见于生命和非生命系统，如树木，铁路，河道网络，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很高的成本效益。因此，要研究这些分支系统形成背后的机制，可以简单地从研究最佳运输系统开始。在过去的几年里，PI和许多其他人开发了用于研究最佳分支运输的几何变分方法，并且还发现其在树叶形成建模中的应用。在这个项目中，主要研究者有兴趣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物体。此外，通过考虑由曲率和运输驱动的表面的几何流，可以使用它们来模拟叶子、花和一些其他分形类型对象的形成。 另一方面，肥皂膜作为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品种中的“肥皂膜”的一般性质，这些品种非常好，但通常包含一些奇异性，在它们的相交同调群中。我们将主要使用几何测度理论作为我们的分析工具。

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.