Coarse Geometry, Discrete Groups, and Operator Algebras

在许多自然的拓扑和几何问题中，人们被引导去研究那些从经典观点来看表现得非常糟糕的空间，例如无限非阿贝尔离散群的酉表示空间。 这样的空间经常被非交换算子代数很好地描述：特别是，鲍姆-康纳斯和粗糙鲍姆-康纳斯代数预言，经典世界和非交换世界之间的良好联系是由更高的指标理论提供的。 这些定理在几何学和拓扑学中有许多应用，例如诺维科夫猜想和正标量曲率度量的存在性。 然而，最近变得明显的是，与离散群和扩展图的某些属性相关的粗糙几何属性是这些属性的障碍：研究者打算系统地研究与这些障碍相关的几何，分析和代数属性，以及因此允许存在的现象。 他还打算应用粗糙几何和非对易几何的相关思想来证明对称空间（具有适当的边界条件）上与椭圆算子相关的经典指标定理。从经典数学的观点来看，几何中出现的许多自然空间具有非常糟糕的性质。 受量子力学的启发，人们试图用非对易的数学结构来描述这些空间--这里的“非对易”意味着人们执行操作的顺序很重要。 如果这样一种“非对易描述”在某种精确意义上是准确的，那么关于几何空间的大量信息就变得可用了。 然而，众所周知，这些描述的某些方面在称为扩展图的空间中失败了--其原因与扩展图在计算机科学和网络理论中有用的原因基本相同。 调查人员计划研究，并使用，“非交换描述”在边界的情况下，特别是空间相关的扩展图和离散的群体，可以用来构建它们。 更好地理解离散群的奇异性质，以及它们与数学和计算机科学其他领域的联系，也是一个中心目标。

In many natural topological and geometric problems, one is led to study spaces that are very badly behaved from a classical point of view, such as the space of unitary representations of an infinite, non-abelian discrete group. Such spaces are quite often described well by noncommutative operator algebras: in particular, the Baum-Connes and coarse Baum-Connes conjectures predict that a good connection between the classical and noncommutative worlds is provided by higher index theory. These conjectures have many applications to geometry and topology, such as to the Novikov conjecture, and to the existence of positive scalar curvature metrics. It has recently become apparent, however, that coarse geometric properties associated to certain quotients of discrete groups and expanding graphs are obstructions to these conjectures: the investigator intends a systematic investigation of the geometric, analytic, and algebraic properties associated to these obstructions, and the phenomena thus allowed to exist. He also intends to apply realted ideas of coarse geometry and noncommuative geometry to prove classical index theorems associated to elliptic operators on symmetric spaces (with suitable boundary conditions).Many natural spaces occurring in geometry have very bad properties from the point of view of classical mathematics. Inspired by quantum mechanics, one tries to describe these spaces using noncommutative mathematical structures - here 'noncommutative' means that the order in which one performs operations matters. If such a 'noncommutative description' is accurate in some precise sense, then a great deal of information about geometric spaces becomes available. It is known, however, that aspects of these descriptions fail for spaces called expanding graphs - for essentially the same reasons that make expanding graphs useful in computer science and the theory of networks. The investigator plans both to study, and to use, 'noncommutative descriptions' in borderline cases, particularly for spaces related to expanding graphs and the discrete groups that can be used to construct them. A better understanding of exotic properties of discrete groups, and thus of their connections to other areas of mathematics and computer science, is also a central goal.