Geometry and Dynamics in Riemannian and Finsler Spaces

本课题可有条件地分为以下（相关）部分：周期度量的研究（包括赋范空间中平面的面积最小化性质，芬斯勒度量的辛填充体积，芬斯勒环面渐近体积增长，积上无共轭点的黎曼度量，无共轭点环面上的拉格朗日系统）；双lipschitz等价与拟等距之间的关系（与一般Penrose平铺和有限呈现群的最有趣的例子，包括同一李群中的协紧格）非交换映射的积、正度量熵流和序列动力学非正曲率几何在算法和动力学中的应用非负弯曲流形的几何（极小曲面的叶状，孤立的平坦全测地线环面）；PI的研究生致力于将有限距离定理推广到非阿贝尔群，通过展开曲面逼近高斯曲率变化较小的嵌入曲面，用规定的雅可比矩阵构造Lipschitz同纯，通过有界子集中元素的共轭乘积生成某些群。项目的第一部分处理周期度量的大规模不变量。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，顺序动力学模型中的稳定性问题，其中对象（例如，物理系统或生态系统）的进化规律受到小扰动的影响；在大时间尺度上理解这种扰动的结果是可取的。奇异空间的现代几何也被应用于源自统计物理的问题（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）和计算问题（例如：如何在数个障碍物之间找到一条最短路径）。项目的最后一部分处理由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物体（例如，一个表面）时，我们处理的是不精确的信息。因此，了解这些信息中的小偏差是否会导致几何对象的关键变化（甚至不存在模型对象）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).