Applications of Non-Commutative Geometry

【摘要】非交换几何试图将几何和拓扑学从经典的黎曼流形和拓扑空间扩展到坐标代数非交换的数学结构的新集合。在这种情况下，大约20年前，P.Baum和a . connes为任何局部紧拓扑群的（约化）C*代数的k理论推测了一个公式。目前这个猜想还没有已知的反例，并且由于许多数学家的工作，这个猜想已经被证明了一些非常有趣的群类（如实李群、p进代数群、幂代数群、离散双曲群、可调群）。还建立了该猜想，当有效时，有许多推论（如Mackey类比，离散级数的Atiyah-Schmid构造，Novikov高签名猜想，稳定的Gromov-Lawson-Rosenberg猜想，Kadison-Kaplansky猜想）。本项目旨在发现和发展表征理论和几何拓扑学中该猜想的进一步推论。分析学是建立在微积分基础上的数学分支。微积分的基本思想（微分和积分）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。拓扑学是几何最基本的形式，是由19世纪和20世纪杰出的数学家黎曼、庞加莱和莱夫谢兹创立的。现代数学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-磁方程是通过分析形成的，但许多含义都是拓扑的。该项目通过使用和应用一种新的分析和拓扑的综合，即“非交换几何”，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".