Topology of Manifolds, C*-Algebras, and Applications

提议摘要：Jonathan Rosenberg教授将研究流形的拓扑和几何，以及C*-代数指标理论及其各种应用。这些应用将包括许多与现代物理学相关的主题，例如正标量曲率流形的分类，与理解大尺度时空几何的可能性有关的问题，以及与弦理论中不同时空拓扑之间的对偶性有关的问题。罗森博格教授密切参与马里兰大学数学研究生课程的监督和改进。他将在代数、分析、几何、拓扑和数学物理方面培训研究生和高级本科生，并将致力于将数学软件整合到本科数学课程中。更详细地说，该提案的一个主要焦点将是通过Yamabe不变量对流形进行分类，以及对正标量曲率的度量进行分类，这些问题涉及微分拓扑，微分几何和分析的相当微妙的混合。另一个重点将是拓扑学和非交换几何在数学物理中的应用，特别是弦理论中的t对偶性。其他主题将包括使用来自C*代数的不变量（特别是卡斯帕罗夫的kk理论）来研究流形的几何和拓扑。例如，来自经典椭圆算子的kk类，如签名算子和Dolbeault算子，将被深入研究。这些研究将加深流形和奇异空间几何的拓扑和解析方法之间的联系。此外，Rosenberg教授将研究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，特别是关于算子代数或几何系统量化的代数。

Proposed abstract:Professor Jonathan Rosenberg will study the topology and geometry of manifolds, as well as C*-algebraic index theory, and various applications. The applications will include a number of topics relevant to modern physics,such as the classification of manifolds of positive scalar curvature,relevant to the problem of understanding the possibilities for thelarge-scale geometry of space-time, and questions relevant to dualities between different space-time topologies in string theory. Professor Rosenberg is closely involved in the supervision and improvement of the mathematics graduate programs at the University of Maryland. He will train graduate students and advanced undergraduate students in algebra, analysis, geometry, topology, and mathematical physics, and will also work toward integration of mathematical software into the undergraduate mathematics curriculum.In more detail, one main focus of the proposal will be the classification of manifolds via their Yamabe invariants, and the classification of metrics of positive scalar curvature, problems which involve a quite subtle blend of differential topology, differential geometry, and analysis. Another focus will be applications of topology and noncommutative geometry to mathematical physics, especially to T-duality in string theory. Other topics will include the use of invariants coming from C*-algebras (especially Kasparov's KK-theory) to study the geometry and topology of manifolds. For example, the KK-classes coming from the classical elliptic operators, such as the signature operator and the Dolbeault operator, will be intensively studied. These studies will deepen the link between topological and analytic approaches to geometry of manifolds and singular spaces. In addition, Professor Rosenberg will study the applications of K-theory to the study of C*-algebras, and various related problems on algebraic K-theory, especially concerning algebras of operators or algebras coming from quantization of geometrical systems.