Coarse Geometry, Discrete Groups, and Operator Algebras

粗略几何、离散群和算子代数

• 批准号: 1229939
• 负责人: Rufus Willett
• 金额: $ 4.53万
• 依托单位: University of Hawaii
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-12-31 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1229939&HistoricalAwards=false
• 关键词: Coarse; Geometry; Discrete; Groups; Operator

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.

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非常糟糕的空间，例如无限的非阿贝尔离散群的酉表示空间。这样的空间通常被非交换算子代数很好地描述：特别是，Baum-Connes猜想和粗糙的Baum-Connes猜想预言了高指数理论提供了经典世界和非交换世界之间的良好联系。这些猜想在几何和拓扑学中有许多应用，例如诺维科夫猜想，以及正标量曲率度量的存在性。然而，最近变得明显的是，与离散群和展开图的某些商相关的粗糙几何性质是这些猜想的障碍：研究者打算系统地研究与这些障碍相关的几何、解析和代数性质，以及因此允许存在的现象。他还打算应用粗糙几何和非交换几何的相关思想来证明对称空间上与椭圆算子相关的经典指数定理（具有合适的边界条件）。从经典数学的观点来看，几何学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，人们试图用非交换的数学结构来描述这些空间——这里的“非交换”意味着执行操作的顺序很重要。如果这种“非交换描述”在某种精确意义上是准确的，那么就可以获得大量关于几何空间的信息。然而，众所周知，这些描述的某些方面不适用于称为展开图的空间，其本质原因与使展开图在计算机科学和网络理论中有用的原因相同。研究者计划在边界情况下研究和使用“非交换描述”，特别是与展开图和可用于构造它们的离散群相关的空间。更好地理解离散群的奇异特性，以及它们与其他数学和计算机科学领域的联系，也是一个中心目标。