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: 1205597
• 负责人: Dmitri Burago
• 金额: $ 14.5万
• 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: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1205597&HistoricalAwards=false
• 关键词: Geometry; Dynamics; Riemannian; Finsler; Spaces

AbstractAward: DMS 1205597, Principal Investigator: Dmitri BuragoThe principal investigator proposes to continue his work on a number of long-term projects in Riemannian and Finsler geometry, dynamical systems of geometric origin, and geometric group theory. The main projects include: geometry of surfaces in normed spaces, ellipticity of surface area functionals, boundary rigidity and related inverse problems, and asymptotic geometry of tori; study of partially hyperbolic diffeomorphisms; conjugation-invariant quasi-norms; "area structures and spaces." The 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 Boundary Rigidity Conjecture, Pu's conjecture, 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, and various generalizations of the E. Hopf Conjecture. In his previous research, the principal investigator was lucky to solve a number of such problems, including Michel's conjecture for nearly flat metrics, non-existence of partially hyperbolic diffeomorphisms on the 3-sphere, the E. Hopf Conjecture on tori without conjugate points, the "Boltzman-Sinaj" problem on the existence of uniform estimates on the number of collisions in hard ball gas models, 2-dimensional cases of Busemann's problem mentioned above, Furstenberg's problem on the existence of bi-Lipschits non-equivalent separated nets and Moser's problem on Jacobian determinants of Lipschitz homeomorphisms, a problem of Hopcroft and Ullman on the complexity of the Split-Find problem, and Shefel's problem on the unboundedness of an immersed complete cylinder with finite total curvature (all from the 40s-70s). Most of the conjectures and directions of the research suggested in this proposal grew from ideas and methods developed by the PI and his collaborators while working on these problems, and the projects described in the proposal continue previous research. The PI has recently started a new research direction, discretization in Riemannian geometry. The new approach is very different from the (classical) PL one, but it addresses both metric and spectral approximations (for differential operators).If one peels off most of confusing mathematical terminology, all parts of the project deal with very practical and even computational understanding of geometric structures. One sends sound waves through the Earth and measures how long it take for them to travel from points to point (sound - because higher frequency waves dissipate, one cannot do X-rays of the Earth, and sound waves are easily detected by seismographs). From this data, how can one figure out what is inside the Earth? The physical analogs of large-scale invariants of periodic metrics are macroscopic properties of periodic media (such as a crystal), and one wants to relate these properties to microscopic characteristics; similarly, study of partially hyperbolic systems, geodesic flows, billiard systems, and geometric complexity may result in better understanding of (stability of) many models in thermodynamics, biology, sociology, and physics, especially when dealing with imprecise data (and in practice, one always deals with imprecise data). If one wants to do a concrete analysis of a complicated geometric object (say, a Riemannian manifold), one has to deal with some finite approximations. Of course in dimension 2 gluing our space out of a bunch of triangles works, but in higher dimension we lose most of its geometric structure. There are more general approximations, but relating their characteristics to geometric properties of actual objects turns out to be a challenging mathematical task.

AbstractAward：DMS 1205597，首席研究员：德米特里BuragoThe首席研究员建议继续他的工作在黎曼和芬斯勒几何，几何起源的动力系统，几何群论的一些长期项目。主要项目包括：赋范空间中的曲面几何，表面积泛函的椭圆性，边界刚性和相关的逆问题，以及环面的渐近几何;部分双曲型同构的研究;共轭不变准范数;“面积结构和空间”。“构成建议核心的项目已经取得了一些重要成果。该提案针对的问题包括Michel的边界刚性猜想，Pu的猜想，Busemann的猜想，即赋范空间中的单位是面积极小化器，部分双曲系统的分类，研究“面积结构”，寻找几何（共轭不变）和“非动力学”（非渐近）不变量的非线性同态，以及E.霍普夫猜想在他以前的研究中，首席研究员很幸运地解决了一些这样的问题，包括Michel猜想的几乎平坦的度量，不存在的部分双曲型同态的3-球，E。无共枕点环面上的Hopf猜想，硬球气体模型中碰撞数一致估计存在性的“Boltzman-Sinaj”问题，上述Busemann问题的二维情形，Furstenberg关于双Lipschits非等价分离网存在性的问题，以及Lipschitz同胚的Jacobian行列式的Moser问题，一个问题的Hopcroft和Ullman的复杂性的分裂找到问题，和Shefel的问题的无界性的浸没完整的圆柱体与有限的总曲率（所有从40年代至70年代）。该提案中建议的大多数研究成果和方向都来自PI及其合作者在研究这些问题时开发的想法和方法，提案中描述的项目延续了以前的研究。PI最近开始了一个新的研究方向，黎曼几何的离散化。 新的方法与（经典的）PL方法有很大的不同，但它同时解决了度量和谱近似（对于微分算子）。如果去掉大部分令人困惑的数学术语，项目的所有部分都涉及对几何结构的非常实用甚至计算的理解。人们发送声波穿过地球，并测量它们从一个点传播到另一个点需要多长时间（声音-因为高频波会消散，人们无法对地球进行X射线检查，而声波很容易被地震仪检测到）。从这些数据中，我们如何才能知道地球内部是什么？ 周期度规的大尺度不变量的物理类比是周期介质的宏观性质（如晶体），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、台球系统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、社会学和物理学中的许多模型的稳定性，特别是在处理不精确数据时（在实践中，人们总是处理不精确数据）。如果要对一个复杂的几何对象（比如黎曼流形）进行具体分析，就必须处理一些有限近似。当然，在二维空间中，将一堆三角形粘在一起是可行的，但在更高的维度中，我们失去了大部分几何结构。还有更一般的近似，但将它们的特性与实际物体的几何特性联系起来是一项具有挑战性的数学任务。