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Foliations: solenoids, regularity and ends

叶状结构：螺线管、规律性和末端

基本信息

批准号：
EP/G006377/1
负责人：
Alexander Clark
金额：
$ 33.8万
依托单位：
University of Leicester
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2008
资助国家：
英国
起止时间：
2008 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FG006377%2F1
关键词：
Foliations solenoids regularity ends

项目摘要

Foliations are extremely important mathematical objects to understand, as they arise naturally in the the solutions to differential equations, level sets of smooth functions and in the geometric study of flows on manifolds.Many of the leading topologists of the past fifty years, including Thurston, Novikov and Milnor, helped develop the theory of foliations to the point that it now is inextricably linked with geometric topology. Foliations have played a significant role in the development of non-commutative geometry, and foliations continue to be an essential element in the modern theories of groupoids and moduli spaces. We aim to significantly increase our understanding of foliations by developing techniques to understand the way the leaves of foliations behave as one approaches infinity . We shall focus in particular on the behaviour in leaves of minimal sets, which can be thought of as the most basic building blocks of a foliation.Ends provide a means to study the asymptotic behaviour of minimal sets of foliations, but there are very few techniques for finding the ends of minimal sets. As an important and highly structured class of minimal sets, solenoids are a natural candidate for just such an analysis. A solenoid is a bundle with a profinite structure group that provided the key we have previously used to unlock many of its important properties.We aim to show that solenoids are prevalent within foliations and thus are ofsignificance for the general understanding of the asymptotics of foliations. At the same time, we shall develop an understanding of the asymptotics of solenoids by developing tools for calculating their ends using the original technique of profinite Cayley graphs. By broadening the scope of these techniques to a wider class ofsolenoids than first considered, we hope to solve the important open question of determining the ends for the general codimension one minimal set. In parallel, we shall undertake a study of the regularity of solenoids as they occur within foliations, showing that the holonomy pseudogroup of a foliation restricted tosolenoidal minimal set must be equicontinuous. Finally, we expect to develop these techniques in enough generality that they could be applied to a much wider class of coarse quasi-isometry invariants. This project will be carried out at the University of Leicester. Due to its broad scope, assistance of a post-doctoral fellow and experts from other universities in the UK and abroad will be required.

叶分是非常重要的数学对象，因为它们自然地出现在微分方程的解、光滑函数的水平集和流形上流动的几何研究中。在过去的50年里，许多顶尖的拓扑学家，包括瑟斯顿、诺维科夫和米尔诺，都帮助发展了叶理理论，使其与几何拓扑学密不可分。叶理在非交换几何的发展中扮演了重要的角色，并且叶理仍然是现代群类群和模空间理论中的重要元素。我们的目标是通过开发技术来理解叶的叶子在接近无穷时的行为方式，从而显着增加我们对叶的理解。我们将特别关注最小集的叶的行为，它可以被认为是叶的最基本的构建块。端点提供了一种研究叶形最小集的渐近行为的方法，但是很少有方法可以找到最小集的端点。作为一个重要的和高度结构化的类最小集，螺线管是一个自然的候选人，只是这样的分析。螺线管是一个具有无限结构组的束，它提供了我们以前用来解锁其许多重要属性的钥匙。我们的目的是表明螺线管是普遍存在于叶理中，因此对叶理渐近的一般理解是有意义的。同时，我们将发展螺线管的渐近性的理解，通过开发工具，计算其端点使用无限的凯利图的原始技术。通过将这些技术的范围扩大到比最初考虑的更广泛的螺线管类别，我们希望解决确定一般余维1最小集的端点的重要开放问题。同时，我们将研究螺线管在叶理中出现的规律性，证明叶理限定螺线管最小集的完整伪群必须是等连续的。最后，我们期望将这些技术发展到足够普遍的程度，使它们可以应用于更广泛的一类粗拟等距不变量。这个项目将在莱斯特大学进行。由于项目范围广泛，需要博士后和来自英国及国外其他大学的专家的协助。