Mathematical Sciences: Operator Algebras, K-Theory and IndexTheory

希格森教授的项目与使用和 算子理论中的指标信息管理 代数一个主要的推动力将是狄拉克算子的方法 关于有边流形的相对K-同调 鲍姆，道格拉斯和泰勒，重点是发展 技术计算一个连接同态， 这个理论的六项正合序列的另一个领域 研究是C*-代数的指标理论， 一致邻域支撑的有界光滑算子 在开流形上的对角线上某些杰出的 这个代数的子代数也将被考虑。 指数的概念渗透到大量的数学中。 基本的表现形式可以在，例如， 卷绕数，用于测量点移动的次数 在一个封闭的平面轨道上绕着一个固定的点旋转， 一般解中任意常数的个数 线性微分方程一个重要的定理，现在有两个 几十年的历史，涉及微分方程的指数行为， 在流形（曲面或高维模拟）上， 关于流形的一般形状，或拓扑结构， 可以用缠绕数来表达。已经 最近发现，结构丰富的数学 称为算子代数的系统提供了一个有利的框架 原指标定理的强有力的扩展。工作 Higson教授的研究将扩大和加强这一框架。

Professor Higson's project has to do with the use and management of index information in the theory of operator algebras. One major thrust will be in the Dirac operator approach to relative K-homology for manifolds with boundary advocated by Baum, Douglas, and Taylor, with an emphasis on developing techniques for computing one of the connecting homomorphisms in the six-term exact sequence for this theory. Another area of investigation is index theory for the C*-algebra generated by bounded smoothing operators supported in a uniform neighborhood of the diagonal on an open manifold. Certain distinguished subalgebras of this algebra will also be considered. The notion of index pervades a great deal of mathematics. Elementary manifestations can be found in, for instance, the winding number that measures the number of times a point moving in a closed planar orbit winds around a fixed point inside, and the number of arbitrary constants in the general solution of a linear differential equation. An important theorem, now a couple of decades old, relates index behavior for differential equations on a manifold (surface or higher-dimensional analogue) to information about the manifold's general shape, or topology, expressible in terms of things like winding numbers. It has been discovered more recently that richly structured mathematical systems called operator algebras provide an auspicious framework for powerful extensions of the original index theorem. The work of Professor Higson will expand and strengthen this framework.