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

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性质，几何研究空间的整体结构，而群论则想知道空间的对称性。度量空间是一个三者都可能感兴趣的对象的例子，这个建议涉及一致离散的度量空间。这样的空间，任何两点之间的距离永远不会小于某个固定的数字(想想黑夜里的恒星)，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，但就像一组恒星开始显示出复杂的形状一样，如果我们从很远的地方观察它(这就是我们观察星系的方式)，如果我们大规模地研究一个离散的度规空间，它就会成为一个有趣的分析对象。格罗莫夫和罗伊为使用这种简单的洞察力提供了一个具体的方案；结果，粗略几何现在是一个既定的工具，当应用于离散的群体时，它特别成功。在有限生成的群中，每个元素都可以用有限字母表写成一个词，短词可以被认为离身份元素近，长词可以被认为远离身份元素。这就产生了一个自然度量，它为群提供了一个均匀的(从每个点看起来都一样)和对称的形状(整个群提供了这个空间的对称性)。这个空间可以通过约化的C*-代数的性质来解析地描述，而这一数学领域中的一些最重要的问题，如Baum-Connes猜想，是由于想要了解这个代数的结构而产生的。这一建议源于我们的发现，即局部类似于粗几何意义上的群的度量空间与群共享许多有趣的分析性质。此外，我们还开发了一个不变量，它允许我们判断度量空间何时与群足够相似。我们的主要新思想，度量空间上的部分平移结构，捕捉了群在自身上的左和右乘法作用的关键组合性质，并提供了一种将这些性质编码到与度量空间相关联的新的C*-代数-部分平移代数中的方法。这一建议的统一策略是发展部分平移结构、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路线。度量空间何时允许一致嵌入到Hilbert空间或群中是一个困难且研究很多的问题。这样的嵌入允许人们控制空间的大规模几何：我们将空间与已知几何的对象进行比较，在第一种情况下，或者在第二种情况下，已知对称。我们将进一步发展我们的技术，以构建新的反例粗略的Baum-Connes猜想，这是一个重要的组织原则，大量研究之间的分析和几何之间的群和度量空间。我们的方法也将提供对瓦莱特猜想的见解，这是几何群论中一个重要的开放问题。这项建议是及时的、雄心勃勃的和要求很高的，并且被放在一个令人兴奋的、快速发展和竞争激烈的数学领域。

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.