Geometry and Dynamics in Riemannian and Finsler Spaces

获奖：DMS 1205597，首席研究员：Dmitri burago首席研究员建议继续他在黎曼和芬斯勒几何，几何起源动力系统和几何群论方面的一些长期项目。主要研究方向包括：赋范空间中曲面的几何、表面积泛函的椭圆性、边界刚性及相关逆问题、环面渐近几何；部分双曲微分同态的研究conjugation-invariant quasi-norms;“区域结构和空间。”构成提案核心的项目已经产生了一些重要成果。该提案旨在解决的问题包括Michel的边界刚性猜想，Pu猜想，Busemann关于赋范空间中的平面是面积最小化的猜想，部分双曲系统的分类，研究“面积结构”，寻找微分同态的几何（共轭不变）和“非动态”（非渐近）不变量，以及E. Hopf猜想的各种推广。在他之前的研究中，首席研究员很幸运地解决了许多这样的问题，包括Michel的近平坦度量猜想，3球上不存在部分双曲微分同态，无共轭点环面上的E. Hopf猜想，关于硬球气体模型中碰撞次数存在一致估计的“Boltzman-Sinaj”问题，上面提到的Busemann问题的二维情况，Furstenberg关于双lipschits非等效分离网存在性的问题，Moser关于Lipschitz同纯态的Jacobian行行式的问题，Hopcroft和Ullman关于Split-Find问题的复杂性的问题，以及Shefel关于有限总曲率浸入式完全圆柱的无界性的问题（都来自40 -70年代）。提案中提出的大多数猜想和研究方向都来自PI和他的合作者在研究这些问题时提出的想法和方法，提案中描述的项目继续了以前的研究。近年来，π开始了一个新的研究方向——黎曼几何中的离散化。新方法与（经典的）PL方法非常不同，但它解决了度量和谱近似（用于微分算子）。如果剥去大多数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性的理解。一种方法是让声波穿过地球，并测量声波从一个点传播到另一个点所需的时间（声音——因为高频波会消散，所以无法对地球进行x射线拍摄，而声波很容易被地震仪探测到）。从这些数据中，我们如何知道地球内部是什么呢？周期度量的大尺度不变量的物理类似物是周期介质（如晶体）的宏观性质，人们希望将这些性质与微观特征联系起来；同样，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更好地理解热力学、生物学、社会学和物理学中的许多模型（稳定性），特别是在处理不精确数据时（实际上，人们总是处理不精确数据）。如果一个人想对一个复杂的几何物体（比如，一个黎曼流形）做一个具体的分析，他必须处理一些有限的近似。当然，在二维空间中，用一堆三角形粘合我们的空间是可行的，但在高维空间中，我们失去了它的大部分几何结构。还有更一般的近似，但将它们的特征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。

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.