Manifolds and C*-Algebraic Index Theory

Johathan M.Rosenberg教授Jonathan Rosenberg将研究流形和类似流形空间的拓扑和几何，以及C*-代数指数理论。这里我们指的是C*-代数理论、椭圆算子指数理论、K-理论、几何和拓扑学的结合。该提案的一个主要焦点将是使用来自C*-代数(特别是Kasparov的KK-理论)的不变量来研究流形的几何和拓扑。例如，来自经典椭圆算子(欧拉特征算子、签名算子和Dolbeault算子)的KK-类将被深入研究。还将注意等变情形，其中有限群作用于所讨论的流形。这些研究将加深拓扑学和解析法与流形理论之间的联系。此外，Rosenberg教授将研究流形的Yamabe不变量分类，以及正标量曲率度量的分类，这些问题涉及到微分拓扑学、微分几何和分析的微妙融合。他还将研究K-理论在C*-代数研究中的应用，以及代数K-理论的各种相关问题，特别是关于算子的代数或来自几何系统量子化的代数。这个项目的背景是使用基于非对易算子代数的新的分析方法来攻击几何和拓扑中的问题。一方面，这些方法是研究几何，特别是高维几何的一种新的工具来源。另一方面，这些方法受到经典物理系统量子化的推动(或多或少是被迫的)。流形的标量曲率性质的研究也受到引力问题的推动。罗森伯格教授将继续为毕业生提供分析、几何和拓扑学方面的培训，并将致力于将数学软件整合到本科数学课程中。

DMS-0103647Johathan M. Rosenberg Professor Jonathan Rosenberg will study the topology and geometry of manifolds and manifold-like spaces, as well as C*-algebraic index theory. By this we mean a combination of C*-algebra theory, index theory of elliptic operators,K-theory, geometry, and topology. One main focus of the proposal will be the use of invariants coming from C*-algebras (especially Kasparov's KK-theory) to study the geometry and topology of manifolds. For example, KK-classes coming from the classical elliptic operators (the Euler characteristicoperator, the signature operator, and the Dolbeault operator)will be intensively studied. Attention will also be paid to the equivariant case, where a finite group acts on the manifold in question. These studies will deepen the link between topological and analytic approaches to manifold theory. In addition, Professor Rosenberg will study the classification of manifolds via their Yamabe invariants, and the classification of metrics of positive scalar curvature, problems which involve a quite subtle blendof differential topology, differential geometry,and analysis. He will also study the applicationsof K-theory to the study of C*-algebras, and various related problems on algebraic K-theory, especially on algebras of operators or on algebras coming from quantization of geometrical systems.The context of this project is the use of new methods in analysis, based on noncommutative operator algebras, for attacking problems in geometry and topology. On the one hand, these methods are useful as a new sourceof tools for studying geometry, especially in high dimensions. And on the other hand, the methods are motivated (and more or less forced) by the quantization of classical physical systems. The study of the scalar curvature properties of manifolds is also motivated by problems in gravitation. Professor Rosenberg will continue to train graduatestudents in analysis, geometry, and topology, and will also work toward integration of mathematical software into the undergraduate mathematics curriculum.