Mathematical Sciences: Operator Algebras, K-Theory and IndexTheory

Higson教授的项目与算子代数理论中索引信息的使用和管理有关。一个主要的推力将是由Baum， Douglas和Taylor倡导的关于具有边界的流形的相对k -同态的Dirac算子方法，重点是为该理论开发计算六项精确序列中的一个连接同态的技术。另一个研究领域是C*-代数的指标理论，这些C*-代数是由开放流形上对角线的一致邻域中支持的有界平滑算子生成的。这个代数的某些特殊的子代数也将被考虑。指数的概念在数学中广泛存在。基本的表现可以在，例如，圈数，测量一个点在一个封闭的平面轨道上绕着一个固定点运动的次数，以及线性微分方程通解中的任意常数的数量中找到。一个重要的定理，现在已经有几十年的历史了，它将流形（曲面或高维模拟）上微分方程的指标行为与流形的一般形状或拓扑信息联系起来，这些信息可以用圈数之类的东西来表示。最近已经发现，称为算子代数的结构丰富的数学系统为原始指标定理的强大扩展提供了一个有利的框架。希格森教授的工作将扩展和加强这一框架。

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.