Applications of Non-Commutative Geometry

分析与拓扑学的相互作用是世纪数学中一个反复出现的主题。“分析”是基于微积分（即微分和积分）的数学，它是从牛顿和莱布尼茨的开创性工作中发展起来的。“拓扑学”是几何学最基本的形式，由世纪和世纪杰出的数学家黎曼、庞加莱和莱夫谢茨创立。黎曼和他的同事罗克陈述并证明了世纪数学的真正瑰宝之一：黎曼-罗克定理。这个著名的定理断言，由分析方法分配给被称为“黎曼曲面上的除数”的数学结构的某些数字实际上是拓扑的。Riemann-Roch定理给出了这些数的拓扑公式。这种黎曼-罗克现象（即某些被解析定义的数实际上是由拓扑公式决定的）在整个世纪数学中以惊人的活力继续着。例子是Lefschetz不动点公式（它导致了A。Weil定理）和椭圆算子的Atiyah-Singer指标定理。本项目的数学延续了这一趋势。30年前，研究人员和A.康纳斯证明了解析不变量（局部紧豪斯多夫拓扑群的约化C*-代数的K-理论）实际上是拓扑的。该猜想引起了广泛的关注，并已成为论文，会议，讲座，博士学位的主题。论文和书籍。这个猜想是不寻常的，因为它跨越了几个不同的数学领域，并揭示了以前被认为是完全无关的问题之间的联系。该项目的主要目标是采取的例子（例如，所有约化的p-adic群），其中猜想现在已知是真实的，并发展的影响。这为众所周知的问题和问题提供了一种新的和不同的方法。同样，该项目使用称为非交换几何的新观点来研究几何拓扑和表示论中的经典问题。非对易几何的观点导致了一些令人吃惊的发现和结果。在约化p-adic群的表示论中，揭示了一种完全出乎意料的几何结构。这大大简化了表示论，并将鲍姆-康纳斯与朗兰兹纲领联系起来。对于几何上产生的Fredholm算子的指数，非交换几何的观点导致了令人惊讶的结论，即像Atiyah-Singer指数公式这样的公式适用于椭圆算子。因此，椭圆性并不是获得这类算子的指数的拓扑公式所需要的要点。

A recurrent theme in nineteenth and twentieth century mathematics has been the interaction of analysis and topology. "Analysis" is mathematics based on calculus (i.e. differentiation and integration), which developed from the pioneering work of Newton and Leibniz. "Topology" is the most basic form of geometry and was founded by such eminent nineteenth century and twentieth century mathematicians as Riemann, Poincare and Lefschetz. Riemann and his co-worker Roch stated and proved one of the true gems of nineteenth century mathematics: the Riemann-Roch theorem. This remarkable theorem asserts that certain numbers assigned by an analytical method to a mathematical structure known as a "divisor on a Riemann surface" are, in fact, topological. The Riemann-Roch theorem gives a topological formula for these numbers. This Riemann-Roch phenomenon (i.e. certain numbers which are analytically defined are in practice determined by a topological formula) has continued with astonishing vigor throughout twentieth century mathematics. Examples are the Lefschetz fixed-point formula (which led to the A. Weil conjectures) and the Atiyah-Singer index theorem for elliptic operators.The mathematics of this project continues this trend. Thirty years ago the investigator and A. Connes conjectured that an analytic invariant (the K-theory of the reduced C*-algebra of a locally compact Hausdorff topological group) is, in fact, topological. The conjecture has drawn wide attention and has been the subject of papers, conferences, lectures, Ph.D. theses, and books. The conjecture is unusual in that it cuts across several different areas of mathematics and reveals connections between problems which previously were thought to be completely unrelated. The main objective of the project is to take the examples (e.g. all reductive p-adic groups) where the conjecture is now known to be true and to develop the implications. This gives a new and different approach to well known problems and issues. Equivalently, the project uses the new point of view known as non-commutative geometry to study classical problems in geometry-topology and representation theory. The non-commutative geometry point of view has led to some startling conjectures and results. In the representation theory of reductive p-adic groups a totally unexpected geometric structure has been revealed. This greatly simplifies the representation theory and links Baum-Connes to the Langlands program. For the index of geometrically-arising Fredholm operators, the non-commutative geometry point of view leads to the surprising conclusion that formulas like the Atiyah-Singer index formula apply well beyond elliptic operators. Thus ellipticity is not the essential point needed to obtain a topological formula for the index of such operators.