Laminations: geometry, harmonic analysis and ergodic theory

提案：DMS-9973086 PI：Alberto Candel摘要：本项目旨在研究层积/叶理领域的几个问题。这些问题之一是与叠片的叶片的几何结构有关。提出者打算研究的一些自然问题是：叠层的表面是什么样的？有多少不同的叶子（例如，准等距类）？什么样的几何不变量可以（不能）用来区分它们？在叶的几何性质方面，还存在着二维分层和度量一致化的问题。特别有趣的是那些有一个欧几里德共形结构，这是经常遇到的问题，低维拓扑和几何群论。本课题的另一部分福尔斯半单群的虚子群的刚性及调和分析。这方面的研究主要集中在具有不变测度的虚子群上。Proposersees旨在发展研究的调和措施，这是在一个合理的方式，以及它们如何适合在现有的身体工作的刚性。该项目的另一部分是关于叶的某些几何性质对其横向或动力学结构以及周围流形的拓扑结构的研究。提议者的研究是在动力系统的定性结构和行为的一般领域。动力学系统被用于模拟许多领域的过程，例如生物体及其形态的进化，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，这些规则可以是事先指定的，也可以是随机的。这可能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）。即使出现在不同的环境中，它们的动态行为和结构也遵循由个体元素和它们相互作用的性质所决定的相似规律。研究者的目的是从几何/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、它们对初始数据的依赖性、可能状态的一般性质以及它们的全局结构有关，所有这些都可以用来理解这些过程的机制。

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.