Geometry and Dynamics in Riemannian and Finsler Spaces

DMS -0103739摘要本项目可有条件地分为以下（相关）部分：定期指标研究（包括赋范空间中平坦面的面积最小化性质、Finsler度量的辛填充体积、Finsler环面的渐近体积增长、乘积上无共枕点的黎曼度量和环面上无共枕点的拉格朗日系统）;双Lipschitz等价与拟等距的关系（与最有趣的情况下，一般彭罗斯tilings和群，包括co-compact格在同一个李群）;产品的非交换地图，流动的正度量熵，和序列动力学;非正曲率几何在算法学和动力学中的应用;非负曲率流形的几何（由极小曲面、孤立平坦全测地环面构成的叶理）; PI的研究生致力于将有限距离定理推广到非阿贝尔群，通过展开曲面来逼近高斯曲率变化较小的嵌入曲面，用指定的Jacobian构造Lipschitz同胚，通过有界子集中元素的共轭乘积来生成某些群。它们的物理类似物是周期性介质（如晶体物质）的宏观性质，问题是如何从微观特征中恢复这些性质，反之亦然。该项目的很大一部分属于几何和动力学之间的边界，特别是几何方法的新应用。例如，序列动力学模型中的稳定性问题，其中对象（例如，物理或生态系统）的演化规律受到小扰动;希望了解这种扰动在大时间尺度上的结果。奇异空间的现代几何也有应用于统计物理学中的问题（例如气体模型中粒子碰撞次数的估计，这个问题可以追溯到玻尔兹曼），以及计算问题（例如：如何在数值上找到几个障碍物之间的最短路径）。该项目的最后一部分涉及的曲率型特征描述的几何对象的稳定性。事实上，每当我们研究一个几何对象（例如，一个表面）时，我们处理的是不精确的信息。因此，重要的是要了解这些信息中的微小偏差是否会导致几何对象的关键变化（甚至是模型对象的不存在）。

Abstract for DMS - 0103739The project can be conditionally divided into the following (related) parts: Study of periodic metrics (including area-minimizing properties of flats in normed spaces, symplectic filling volumes for Finslermetrics, asymptotic volume growth of Finsler tori, Riemannian metrics without conjugate points on products, and Lagrangian systems on tori without conjugate points); Relationship between bi-Lipschitz equivalence and quasi-isometries (with the most intriguing cases of general Penrose tilings and finitely presented groups, including co-compact lattices in the same Lie group); Products of non-commuting maps, flows of positive metric entropy, and sequential dynamics; Applications of geometry of non-positive curvature to algorithmics and dynamics; Geometry of non-negatively-curved manifolds (foliations by minimal surfaces, isolated flat totally geodesic tori); the PI's graduate students work on generalizations of the Finite Distance Theorem to non-Abelian groups, approximations of embedded surfaces with small variation of Gaussian curvature by developing surfaces, constructing Lipschitz homeomorphisms with prescribed Jacobians, generating certain groups by products of conjugates of elements from a bounded subset.The first part of the project deals with large-scale invariants of periodic metrics. Their physical analogs are macroscopic properties of periodic media (such as a crystal substance), and the problem is to understand how such properties can be recovered from microscopic characteristics and vice versa. A large part of the project belongs to a borderline between geometry and dynamics, and in particular new applications of geometric methods. For instance, problems of stability in sequential dynamics model situations where the laws of evolution of an object (for instance, a physical or an ecological system) are subject to small perturbations; it is desirable to understand the result of such perturbations in the large time scale. There are also applications of modern geometry of singular spaces to problems originated from statistical physics (such as estimates on the number of collisions of particles in gas models, a problem that goes back to Boltzmann), and to computational problems (such as: how to numerically find a shortest path betweenaround several obstacles). The last part of the project deals with stability of geometric objects described by curvature-type characteristics. Indeed, whenever we study a geometric object (for instance, a surface), we deal with imprecise information. Thus it is important to understand whether small deviations in this information can result in crucial changes for the geometric object (or even a non-existence of a model object).