Applications of Non-Commutative Geometry

非交换几何试图将几何学和拓扑学从经典的黎曼流形和拓扑空间扩展到一个新的数学结构，其坐标代数是非交换的。 在此背景下，大约20年前，P. Baum和A. Connes给出了任何局部紧拓扑群的（约化）C * 代数的K-理论的公式。在目前的时间没有反例是已知的猜想和-由于工作的许多数学家-猜想已被证明为几个非常有趣的类的群体（例如真实的李群，p进代数群，adelic代数群，离散双曲群，顺从的群体）。还建立了猜想，当有效时，有许多推论（例如Mackey类比，Atiyah-Schmid离散级数的构造，Novikov高阶签名猜想，稳定Gromov-Lawson-Rosenberg猜想，Kadison-Kaplansky猜想）。该项目旨在发现和发展表示论和几何拓扑学中猜想的进一步推论。分析是数学的分支，以微积分为基础。微积分（微分和积分）的基本思想是由牛顿和莱布尼茨提出的，并在他们那个时代的科学革命中发挥了核心作用。拓扑学是几何学最基本的形式，由19和20世纪杰出的数学家黎曼、庞加莱和莱夫谢茨创立。现代数学中的一个主要主题是分析和拓扑学之间的相互作用。例如，麦克斯韦的电磁方程是通过分析来表述的，但其中许多含义是拓扑的。这个项目通过使用和应用一种新的被称为“非交换几何”的分析和拓扑的综合来继续拓扑和分析的相互作用。

AbstractBaumNon-commutative geometry seeks to extend geometry and topology from the classical setting of Riemannian manifolds and topological spaces to a new setting of mathematical structures whose coordinate algebras are non-commutative. In this context, approximately twenty years ago, P.Baum and A.Connes conjectured a formula for the K-theory of the (reduced) C* algebra of any locally compact topological group. At the present time no counter-example is known to the conjecture and-due to the work of many mathematicians-the conjecture has been proved for several very interesting classes of groups (e.g. real Lie groups, p-adic algebraic groups, adelic algebraic groups, discrete hyperbolic groups, amenable groups). Also established is that the conjecture, when valid, has many corollaries (e.g. Mackey analogy, Atiyah-Schmid construction of the discrete series, Novikov higher signature conjecture, stable Gromov-Lawson-Rosenberg conjecture, Kadison-Kaplansky conjecture). This project aims to discover and develop further corollaries of the conjecture in representation theory and in geometry-topology.Analysis is the branch of mathematics based on calculus. The fundamental ideas of calculus (differentiation and integration) were introduced by Newton and Leibniz and played a central role in the scientific revolution of their era. Topology is the most basic form of geometry and was founded by such eminent nineteenth and twentieth century mathematicians as Riemann, Poincare and Lefschetz. A major theme in modern mathematics has been the interplay between analysis and topology. For example, Maxwell's equations for electricity-magnetism are formulated via analysis, but many of the implications are topological. This project continues the interaction of topology and analysis by using and applying a new synthesis of analysis and topology known as "non-commutative geometry".