Foliations: solenoids, regularity and ends

叶理是非常重要的数学对象来理解，因为它们自然出现在解决微分方程，水平集的光滑函数和几何研究的流动流形。许多领先的拓扑学家在过去的50年，包括瑟斯顿，诺维科夫和米尔诺，帮助发展理论的叶理的一点，它现在是不可分割的联系与几何拓扑。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胚和模空间理论中的一个基本元素。我们的目标是显着增加我们的理解，通过开发技术，了解的方式，叶理的行为，因为一个接近无穷大的叶理。我们将特别关注最小集的叶的行为，它可以被认为是最基本的构建块的foliation.Ends提供了一种手段来研究最小集的foliations.Ends渐近行为的最小集，但有很少的技术找到最小集的结束。作为一个重要的和高度结构化的最小集类，类是一个自然的候选人，只是这样的分析。螺线管是一个具有profinite结构群的丛，它提供了我们以前用来解开它的许多重要性质的钥匙。我们的目的是表明，在叶理中，渐近性是普遍存在的，因此对于一般理解叶理的渐近性是有意义的。与此同时，我们将通过开发工具来计算它们的端点，使用原始的profinite凯莱图技术，来理解递归的渐近性。通过扩大这些技术的范围，以更广泛的类oferies比第一次考虑，我们希望解决重要的开放问题，确定一般余维一个最小集的结束。与此同时，我们将进行一项研究的规律性，因为它们发生在叶理，表明holonomy伪群的叶理限制到solenoidal最小集必须是等度连续的。最后，我们希望开发这些技术在足够的一般性，它们可以应用到更广泛的一类粗准等距不变量。该项目将在莱斯特大学进行。由于其广泛的范围，博士后研究员和来自英国和国外其他大学的专家的援助将是必需的。

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.