Geometry and Dynamics in Riemannian and Finsler Spaces

DMS-0103739该项目可有条件地分为下列(相关)部分：周期度量(包括赋范空间中平坦的面积最小化性质、Finsler度量的辛填充体积、Finsler环面的渐近体积增长、乘积上无共轭点的黎曼度量以及环面上无共轭点的拉格朗日系统)；双Lipschitz等价与准等距之间的关系(具有一般PenRose平铺和有限表示群的最有趣的情况，包括同一Lie群中的余紧格)；非交换映射的乘积、正向度量流和序列；非正曲率几何在算法和动力学中的应用；非负曲线流形的几何(极小曲面的叶状，孤立的平坦全测地环面)；PI的研究生致力于将有限距离定理推广到非阿贝尔群，通过开发曲面来逼近具有小的高斯曲率变化的嵌入曲面，用指定的雅可比构造Lipschitz同胚映射，通过从有界子集生成元素的共轭乘积来生成某些群。项目的第一部分涉及周期度量的大规模不变量。它们的物理类比是周期性介质(如晶体物质)的宏观性质，问题是如何从微观特征恢复这些性质，反之亦然。该项目的很大一部分属于几何学和动力学之间的分界线，特别是几何方法的新应用。例如，在顺序动力学模型情况下的稳定性问题，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应用于源自统计物理的问题(例如对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)，也应用于计算问题(例如：如何从数值上找到绕过几个障碍物之间的最短路径)。该项目的最后部分涉及由曲率类型特征描述的几何对象的稳定性。事实上，每当我们研究几何对象(例如，曲面)时，我们都要处理不精确的信息。因此，了解此信息中的微小偏差是否会导致几何对象(或者甚至不存在模型对象)的重要变化是很重要的。

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).