Laminations: geometry, harmonic analysis and ergodic theory

摘要：本课题拟研究火焰/叶化领域的几个问题。其中一个问题与层压叶片的几何结构有关。提议者想要研究的一些自然问题是：层压的叶子是什么样子的？有多少不同的叶子(例如？准等距类)有吗？哪些几何不变量可以（或不能）用来区分它们？仍然关注叶片的几何形状，存在关于叶片的二维分层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，这在低维拓扑和几何群论问题中经常遇到。本课题的另一部分是半单群虚子群的刚性及其调和分析。这一领域的研究主要关注具有不变测度的虚子群。该提案旨在发展谐波测度的研究，谐波测度以一种合理的方式是非不变的，以及它们如何适应现有的刚性工作。该项目的另一部分涉及研究叶状叶的某些几何特性对其横向或动态结构以及环境流形的拓扑结构的影响。作者的研究方向是动力系统的定性结构和行为。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。其他令人感兴趣的系统来自于根据一组规则从有限数量的数据进化而来的过程，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着由单个元素及其相互作用的性质所决定的类似规律。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.