Laminations: geometry, harmonic analysis and ergodic theory

叠片：几何、调和分析和遍历理论

• 批准号: 9973086
• 负责人: Alberto Candel
• 金额: $ 6.76万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2001-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9973086&HistoricalAwards=false
• 关键词: Laminations; geometry; harmonic; analysis; ergodic

Proposal: DMS-9973086PI: Alberto CandelAbstract: This project proposes to study several problems in the area oflaminations/foliations. One of these problems is concerned with thegeometric structure of the leaves of a lamination. Some naturalquestions that the proposer intends to study are: what do the leavesof a lamination look like? how many different leaves(e.g. quasi-isometry classes) are there? what geometric invariants canbe (and cannot be) used to distinguish them? Still concerned with thegeometry of the leaves, there are questions about two dimensionallaminations and metric uniformization of their leaves. Especiallyinteresting are those which have a Euclidean conformal structure,which are often encountered in problems of low dimensional topologyand geometric group theory. Other part of this project falls in thearea of rigidity of virtual subgroups of semisimple groups andharmonic analysis on them. Research in this area has been concernedwith virtual subgroups which have an invariant measure. The proposerseeks to develop the study of harmonic measures, which are noninvariant in a reasonable way, and how they fit in the existing bodyof work on rigidity. Other part of the project concerns the study thatcertain geometric properties of the leaves of a foliation have on itstransverse or dynamical structure and on the topological structure ofthe ambient manifold.The research of the proposer is in the general area of qualitativestructure and behavior of dynamical systems. Dynamical systems areused to model processes in many areas, for example the evolution ofliving organisms and their morphology, the weather or other physicalor chemical processes. Other systems of interest come from processeswhich evolve from a finite amount of data according to some set ofrules, either specified before hand or of a random nature. This couldbe neural networks in the brain, systems of digital processors in acomputer, branching processes (as in nuclear fission, evolution andgrowth of cells, etcetera). Even when appearing under differentcircumstances, their dynamic behavior and structure follows similarlaws dictated by the individual elements and the nature of theirinteraction. The purpose of the researcher is to study and classify,from a geometric/qualitative viewpoint, invariants of these systemsrelating to their long time asymptotic behavior, their dependence onthe initial data, the generic properties of the possible states, andtheir global structure, all which can be used to understand themechanisms underlying such processes.

建议：DMS-9973086PI：Alberto Candel摘要：本项目建议研究火焰/叶理领域的几个问题。其中一个问题与叠层叶片的几何结构有关。提出者打算研究的一些自然问题是：薄层的叶子是什么样子的？有多少种不同的叶子(如准等距类)？哪些几何不变量可以(以及不能)用来区分它们？仍然与叶的几何有关，存在关于其叶的二维同构和度量均匀化的问题。特别有趣的是那些具有欧几里得共形结构的结构，这是在低维拓扑和几何群论中经常遇到的问题。本项目的其他部分属于半单群的虚子群的刚性及其调和分析的领域。这一领域的研究一直关注于具有不变测度的虚拟子群。作者致力于发展调和测度的研究，它以一种合理的方式是不变的，以及它们如何与现有的刚性工作相适应。该项目的另一部分是关于叶层的某些几何性质在其横向或动力结构上以及在环境流形的拓扑结构上的研究。动力系统被用来模拟许多领域的过程，例如生物体的进化及其形态、天气或其他物理或化学过程。其他感兴趣的系统来自根据一些规则集从有限数量的数据演变而来的过程，这些规则要么是事先指定的，要么是随机的。这可以是大脑中的神经网络、计算机中的数字处理器系统、分支过程(如核裂变、细胞的进化和生长等)。即使在不同的情况下出现，它们的动态行为和结构也遵循由个别元素和它们相互作用的性质所决定的相似规律。研究人员的目的是从几何/定性的角度研究和分类这些系统的不变量，这些不变量与它们的长时间渐近行为有关，它们对初始数据的依赖，可能状态的一般性质，以及它们的全局结构，所有这些都可以用来理解这些过程背后的机制。