RUI: Surfaces and their horizons, geometric structures, and pseudogroups

RUI：曲面及其视界、几何结构和伪群

• 批准号: 0205825
• 负责人: Alberto Candel
• 金额: $ 9.9万
• 依托单位: The University Corporation, Northridge
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-10-01 至 2006-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0205825&HistoricalAwards=false
• 关键词: RUI; Surfaces; their; horizons; geometric

The proposed research is in the area of geometry anddynamics. The objective is to understand the asymptotic behaviorof orbits of dynamical systems. One specific problem deals withsurface laminations in hyperbolic three-manifolds. Rather thanstudying recurrence phenomena taking place in a compact space aglobal approach is taken. The geometry of hyperbolic space has awell defined visual boundary representing the many possible waysof diverging to infinity. Moreover, the ambient hyperbolicgeometry affects the geometry of the leaves of the laminationunder consideration. The basic problem is then to study how thesegeometries relate with respect to their approach to the visualboundaries, and to examine the influence that the ambienthyperbolic geometry exerts on the leaves. Within this program itis also natural to consider laminations whose leaves havestronger geometric or analytic properties, or both, as forexample when they satisfy the minimal surface differentialequation. Such hypothesis allows for the use of analytical andprobabilistic tools, and more precise information can beobtained. Laminations whose leaves are minimal surfaces are alsorelevant to understand the geometry of other three-manifolds aswell. They appear to play an important role in the topologicalhyperbolization conjecture for three-manifolds, as is known thatnon-hyperbolic three manifolds have a lamination by minimalsurfaces. This proposal also includes problems in the area ofrigidity of actions of semisimple groups. The main focus is inthe so-called Gromov's centralizer theorem, a major tool inunderstanding the symmetries of geometric structures onmanifolds. Other questions relating to the structure ofpseudogroups of transformations are also proposed.Dynamical systems are used to model processes in many areas, forexample the weather, physical or chemical processes, and theevolution of living organisms and their morphology. They are alsoused for modeling processes which evolve from a finite amount ofdata according to some set of rules, either specified before handor of a random nature, as neural networks in the brain or systemsof digital processors in a computer. The objective of theproposed research is to understand the qualitative structure andasymptotic behavior of certain dynamical systems. One of thetopics in this proposal is to study the behavior oftwo-dimensional systems evolving in a three-dimensional space,and specifically the interaction of the features of thesurrounding space and the geometry and asymptotic behavior oftheir trajectories. This can be approached in a variety of ways:by purely geometric means, by studying certain differentialequations that define their orbits, or via a probabilisticapproach. Understanding basic features of these dynamical systemsand these spaces, while having intrinsic geometric interest andbeauty, could be relevant, for example, in areas like partialdifferential equations, solid state physics, structure ofcrystals and quasi-crystals and their defects, statisticalmechanics, computation and algorithms.

拟议的研究是在几何学和动力学领域。其目的是了解动力系统轨道的渐近行为。一个特殊的问题涉及双曲三维流形中的表面叠层。不是研究在紧凑空间中发生的重复现象，而是采取全局方法。双曲空间的几何学有一个明确的视觉边界，代表了发散到无穷远的许多可能的方式。此外，环境双曲几何影响所考虑的层合板的叶片的几何形状。然后，基本的问题是研究这些几何如何与它们接近视觉边界的方法相关联，并检查双曲几何对树叶的影响。在本程序中，也很自然地考虑其叶片具有更强的几何或解析性质，或两者兼有的层合板，例如当它们满足最小曲面微分方程式时。这样的假设允许使用分析和概率工具，并且可以获得更精确的信息。叶是极小曲面的薄片也与理解其他三种流形的几何有关。它们似乎在三维流形的拓扑双曲化猜想中起着重要的作用，因为众所周知，非双曲的三个流形具有极小曲面的叠层。这一建议还包括半单群的作用的刚性方面的问题。主要的焦点是所谓的格罗莫夫中心化子定理，它是理解流形上几何结构对称性的主要工具。动力学系统被用来模拟许多领域的过程，例如天气、物理或化学过程，以及生物体的进化和它们的形态。它们也被用于建模过程，这些过程是根据某种规则集从有限数量的数据演变而来的，这些规则要么是事先指定的，要么是随机性质的，比如大脑中的神经网络或计算机中的数字处理器系统。所提出的研究目的是了解某些动力系统的定性结构和渐近行为。该方案的主要内容之一是研究二维系统在三维空间中的演化行为，特别是研究环绕空间的特征与其轨迹的几何和渐近行为之间的相互作用。这可以用各种方法来解决：通过纯几何方法，通过研究定义其轨道的某些微分方程式，或通过概率方法。了解这些动力系统和这些空间的基本特征，虽然具有内在的几何兴趣和美感，但可能与诸如偏微分方程组、固体物理、晶体和准晶体的结构及其缺陷、统计力学、计算和算法等领域有关。