Geometry and Dynamics in Riemannian and Finsler Spaces

黎曼空间和芬斯勒空间中的几何和动力学

• 批准号: 0905838
• 负责人: Dmitri Burago
• 金额: $ 14.93万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0905838&HistoricalAwards=false
• 关键词: Geometry; Dynamics; Riemannian; Finsler; Spaces

The main projects of the proposed researchinclude: boundary rigidity and related inverse problems, minimal fillings, asymptotic geometry of tori and area-minimizing surfaces in normed spaces; study of partially hyperbolic diffeomorphisms; conjugation-invariant quasi-norms; ``area structures and spaces". They are aiming at solving a number of long-standing and important problems, and formulating new problems and new directions of research. The four projects that form the core of the proposal have already yielded a number of important results.Among the problems the proposal is aimed at are Michel's Conjecture that simple metrics are boundary rigid, Pu's conjecture on the filling area of the circle, Busemann's Conjecture that flats in normed spaces are area-minimizers, classifications of partially-hyperbolic systems, studying ``area structures", finding geometric (conjugation-invariant) and ``non-dynamical" (not asymptotic) invariants of diffeomorphisms preserving certain structures, and various generalizations of the E. Hopf Conjecture.Most of the conjectures and directions of research suggested in the proposal grew from ideas and methods developed by the PI in his previous research, and projects described in the proposal continue the research resulted in solving them.Many topics of the proposed research have rather feasible relation to applied science. The Boundary Rigidity Problem and related Inverse Problems are motivated by important problems in geophysics and medical imaging.To visualize that, imagine that one wants to find out what the Earth is made of.More generally, one wants to find out what is inside a solid body made of different materials (in other words, properties of the medium change from point to point). The speed of sound depends on the material. One can "tap" at somepoints of the surface of the body and "listen when the sound gets to other points".The question is if this information is enough to determine what is inside. The proposed research already resulted in the first result of this kind for a reasonably general case (with the restriction that the properties of the material do not change to much from point to point), and there is a cautions hope to handle the general case.The physical analogs of large-scale invariants of periodic metrics are macroscopic properties of periodic media (such as a crystal substance), and one wants to relate these properties to microscopic characteristics; similarly, study of partially hyperbolic systems, geodesic flows, and geometric complexity may result in better understanding of (stability) of certain models in thermodynamics, biology, sociology, and physics, especially when dealing with imprecise data.

拟开展的主要研究项目包括：赋范空间中的边界刚性及相关的反问题、极小填充、环面和面积极小曲面的渐近几何;部分双曲双同态的研究;共轭不变拟范数;"面积结构与空间”。它们旨在解决一些长期存在的重要问题，并制定新问题和新的研究方向。作为该提案核心的四个项目已经取得了一些重要成果。该提案所针对的问题包括：简单度量是边界刚性的米歇尔猜想、关于圆的填充面积的浦氏猜想、赋范空间中的平坦是面积极小的布斯曼猜想、部分双曲系统的分类、研究"面积结构”、寻找保持某些结构的非同态的几何（共轭不变）和"非动力学”（非渐近）不变量，以及E.霍普夫猜想（Hopf Conjecture）。提案中提出的大部分猜想和研究方向都是从PI在以前的研究中开发的想法和方法中发展出来的，提案中描述的项目继续进行解决这些问题的研究。提案中提出的许多研究课题与应用科学有着相当可行的关系。边界刚度问题和相关的逆问题是由地球物理学和医学成像中的重要问题激发的。为了将其可视化，想象一下人们想要知道地球是由什么组成的。更一般地说，人们想要知道由不同材料制成的固体内部有什么（换句话说，介质的属性从一个点到另一个点发生变化）。声速取决于材料。一个人可以“轻敲”身体表面的某些点，“当声音到达其他点时听”，问题是这些信息是否足以确定身体内部是什么。所提出的研究已经在一个合理的一般情况下产生了第一个这种结果周期度规的大尺度不变量的物理类似物是周期介质的宏观性质（如晶体物质），人们希望将这些性质与微观特征联系起来;类似地，对部分双曲系统、测地线流和几何复杂性的研究可能会导致更好地理解热力学、生物学、社会学和物理学中某些模型的稳定性，特别是在处理不精确的数据时。