权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Classifying spaces for proper actions and almost-flat manifolds

对空间进行分类以实现正确的操作和几乎平坦的流形

基本信息

批准号：
EP/N033787/1
负责人：
Nansen Petrosyan
金额：
$ 12.63万
依托单位：
University of Southampton
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2016
资助国家：
英国
起止时间：
2016 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FN033787%2F1
关键词：
Classifying spaces proper actions almost

项目摘要

In this research, we will combine techniques from Geometric Group Theory, Topololgy, and Geometry to work on two objectives. In the last twenty years, non-positively curved spaces and groups have been at the forefront of Geometric Group Theory and Topology. Their importance is underlined by I. Agol's breakthrough solution of the Virtual Haken Conjecture of Thurston using the machinery of non-positively curved cube complexes developed by D. Wise. Also, in the last decade, the Baum-Connes and the Farrell-Jones Conjectures have been verified for many (non-positively curved) classes of groups, paving the way for computations in algebraic K- and L-theories via their classifying spaces. These conjectures connect many different fields of mathematics and have far reaching applications in Topology, Analysis, and Algebra. The time is therefore right to investigate (finiteness) properties of such groups and to construct models for classifying spaces for proper actions with geometric properties that are suitable for computations. Our first objective is to construct such models for classifying spaces of proper actions for some important classes of groups such as Coxeter groups and the outer automorphism group of right-angled Artin groups, and to investigate Brown's conjecture.Our second objective is on almost-flat manifolds. These manifolds are a generalisation of flat manifolds introduced by M. Gromov. They occur naturally in the study of Riemannian manifolds with negative sectional curvature and play a key role in the study of collapsing manifolds with uniformly bounded sectional curvature. The characteristic properties of these manifolds that we will investigate such as Spin structures and cobordisms play an integral part in modern manifold theory. Spin structures have many applications in Quantum Field Theory and in Mathematical Physics. In particular, the existence of a Spin structure on a smooth orientable manifold allows one to define spinor fields and a Dirac operator which can be thought of as the square root of the Laplacian. Dirac operator is essential in describing the behaviour of fermions in Particle Physics. It is also an important invariant in Pure Mathematics arising in Atiyah-Singer Index Theorem, Connes's Noncommutative Differential Geometry, the Schrodinger-Lichnerowicz formula, Kostant's cubic Dirac operator, and many other areas. The methods by which we propose to study almost-flat manifolds arise from the interactions between Geometry/Topology and Group Theory. This is largely due to the fact that the topology of these manifolds is completely classified by their fundamental groups.

在本研究中，我们将结合几何群论、拓扑学和几何学的联合收割机技术来实现两个目标。在过去的二十年里，非正曲空间和群一直处于几何群论和拓扑学的前沿。我强调了它们的重要性。Agol利用D。睿的此外，在过去的十年中，鲍姆-康纳斯猜想和法雷尔-琼斯猜想已经在许多（非正曲）群类中得到了验证，为代数K-和L-理论通过其分类空间的计算铺平了道路。这些代数连接了许多不同的数学领域，在拓扑学、分析学和代数学中有着深远的应用。因此，现在是时候调查（有限性）性质，这些群体和建设模型的分类空间适当的行动与几何性质，是适合于计算。我们的第一个目标是为一些重要的群类（如Coxeter群和直角Artin群的外自同构群）构造这样的模型来分类真作用空间，并研究Brown猜想;第二个目标是在几乎平坦流形上。这些流形是M.格罗莫夫它们自然地出现在具有负截面曲率的黎曼流形的研究中，并且在具有一致有界截面曲率的坍缩流形的研究中起着关键作用。我们将研究的这些流形的特征性质，如自旋结构和配边，在现代流形理论中起着不可或缺的作用。自旋结构在量子场论和数学物理中有许多应用。特别是，在光滑可定向流形上存在自旋结构允许定义旋量场和狄拉克算子，可以认为是拉普拉斯算子的平方根。狄拉克算符是粒子物理中描述费米子行为的基本算符。它也是纯数学中的一个重要不变量，出现在Atiyah-Singer指标定理、Connes的非交换微分几何、Schrodinger-Lichnerowicz公式、Kostant的三次狄拉克算子和许多其他领域。我们提出的研究几乎平坦流形的方法来自几何/拓扑和群论之间的相互作用。这在很大程度上是由于这些流形的拓扑结构完全由它们的基本群分类。