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Analysis and geometry of metric spaces with applications in geometric group theory and topology.

度量空间的分析和几何及其在几何群论和拓扑中的应用。

基本信息

批准号：
EP/F031947/1
负责人：
Jacek Brodzki
金额：
$ 45.57万
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：
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%2FF031947%2F1
关键词：
Analysis geometry metric spaces applications

项目摘要

Analysis, geometry and group theory are three of the main classical areas of mathematics. Analysis studies local properties of a space, geometry is concerned with its overall structure, while group theory wants to know the space's symmetries. A metric space is an example of an object which is potentially of interest to all three, and this proposal is concerned with uniformly discrete metric spaces. Such spaces, where the distance between any two points is never smaller than some fixed number (think of stars on a dark night) do not seem to have enough local structure to make them interesting from the point of view of analysis, but just as a collection of stars begins to display an intricate shape if we look at it from a large distance (this is how we observe galaxies), a discrete metric space becomes an interesting analytic object if we study it on the large scale. Gromov and Roe provided a concrete scheme for using this simple insight; the result, coarse geometry, is now an established tool, and has been particularly successful when applied to discrete groups. In a finitely generated group, where every element can be written as a word using a finite alphabet, short words can be regarded as being close to the identity element, and long words far away from it. This leads to a natural metric, which gives the group a shape that is homogeneous (looks the same from every point) and symmetric (the whole group provides symmetries of this space). This space can be described analytically through the properties of the reduced C*-algebra, and some of the most important questions in this area of mathematics, like the Baum-Connes conjecture, arise from desire to understand the structure of this algebra. This proposal arises from our discovery that metric spaces which locally resemble groups in the coarse-geometric sense share with groups a lot of interesting analytic properties. Moreover, we have developed an invariant that allows us to say when a metric space is sufficiently similar to a group. Our main new idea, the partial translation structure on a metric space, captures the key combinatorial properties of the left and right mulitplication action of a group on itself and provides a method of encoding those properties in a new C*-algebra, the partial translation algebra, that we associate with a metric space. A unifying strategy of this proposal is the development of partial translation structures, partial translation C*-algebras and our invariant to provide new routes of attack on important outstanding problems. A difficult and much studied question is when a metric space admits a uniform embedding into a Hilbert space or a group. Such an embedding allows one to control the large scale geometry of a space: we compare the space with an object of known geometry, in the first case, or known symmetry, in the second. We will further develop our techniques to construct new counterexamples to the coarse Baum-Connes conjecture, which is an important organising principle for a large body of research on the interface between the analysis and geometry of groups and metric spaces. Our approach will also provide insights into the Valette conjecture, which is an important open question in geometric group theory. This proposal is timely, ambitious and demanding, and is placed in an exciting, rapidly developing and competitive area of mathematics.

分析、几何和群论是数学的三个主要经典领域。分析研究空间的局部属性，几何学关注其整体结构，而群论则想了解空间的对称性。度量空间是所有三个人都可能感兴趣的对象的一个示例，并且该提案涉及均匀离散的度量空间。在这样的空间中，任何两点之间的距离永远不会小于某个固定数字（想象一下黑夜中的星星），从分析的角度来看，似乎没有足够的局部结构使它们变得有趣，但正如如果我们从很远的距离观察星星的集合开始显示出复杂的形状一样（这就是我们观察星系的方式），如果我们大规模地研究它，离散度量空间就会成为一个有趣的分析对象。格罗莫夫和罗伊提供了一个利用这一简单见解的具体方案；结果，粗几何图形现已成为一种成熟的工具，并且在应用于离散组时特别成功。在有限生成群中，每个元素都可以使用有限字母表写成一个单词，短单词可以被视为靠近单位元素，而长单词则可以被视为远离单位元素。这导致了一个自然的度量，它赋予该群一个均匀的（从每个点看起来都相同）和对称的形状（整个群提供了这个空间的对称性）。这个空间可以通过简化的 C* 代数的性质来分析地描述，而这个数学领域中的一些最重要的问题，比如鲍姆-康内斯猜想，都是出于理解这个代数结构的愿望而产生的。这个提议源于我们的发现，即在粗略几何意义上局部类似于群的度量空间与群共享许多有趣的分析属性。此外，我们还开发了一个不变量，可以让我们判断度量空间何时与群足够相似。我们的主要新想法是度量空间上的部分平移结构，它捕获了群对其自身的左右乘法作用的关键组合属性，并提供了一种在新的 C* 代数（部分平移代数）中对这些属性进行编码的方法，我们将其与度量空间相关联。该提案的统一策略是开发部分平移结构、部分平移 C* 代数和我们的不变量，为重要的突出问题提供新的攻击途径。一个困难且经过大量研究的问题是度量空间何时允许均匀嵌入到希尔伯特空间或群中。这种嵌入允许人们控制空间的大规模几何形状：在第一种情况下，我们将空间与已知几何形状的对象进行比较，在第二种情况下，我们将空间与已知对称性的对象进行比较。我们将进一步开发我们的技术来构造粗鲍姆-康尼斯猜想的新反例，这是关于群和度量空间的分析和几何之间的接口的大量研究的重要组织原则。我们的方法还将提供对瓦莱特猜想的见解，这是几何群论中一个重要的悬而未决的问题。这个提议是及时的、雄心勃勃的和要求很高的，并且被置于一个令人兴奋、快速发展和竞争激烈的数学领域。