Mathematical Sciences: Noncommutative Geometry

摘要Wodzicki在这个项目中，Wodzicki将研究K-理论和算子理想的循环同调，以及Hilbert空间上系数为理想的有界算子代数的Hochschild同调。特别是，他将学习：行列式，调节器，更高的解析和拓扑指数不变量，希尔伯特空间上有界线性算子的交换子结构，以及与代数簇的K-理论的联系。这一主题的结合自然是首席调查员先前工作的结果。这种方法结合了数学的几个领域，特别关注具有解析和几何意义的(非交换)代数的同调研究，以及适当的同调和同伦工具的发展。该项目触及了现代数学发展的基本方向，并解决了数学中具有根本重要性和趣味性的问题。因此，这是一项“纯粹的研究”。然而，它可能在未来在高能和核物理等领域产生相当大的影响。目前的项目是首席研究员过去15年研究计划的自然延续和发展。他在这个项目中获得的早期结果已经在几位物理学家最近的工作中得到了应用(特别是那些与CERN--位于日内瓦的欧洲粒子物理实验室有关的物理学家)。

ABSTRACT Wodzicki In this project Wodzicki shall study K-theory and cyclic homology of operator ideals, and also the Hochschild homology of the algebra of bounded operators on a Hilbert space with coefficients in an ideal. In particular, he will study: determinants, regulators, higher analytic and topological index invariants, the commutator structure of bounded linear operators on a Hilbert space, and a connection with the K-theory of algebraic varieties. This combination of topics naturally results from the previous work of the principal investigator. The approach joins together several areas of mathematics with a special focus on homological study of (noncommutative) algebras of analytic and geometric significance and on the development of suitable homological and homotopical tools. The project touches upon the fundamental directions in which modern mathematics is developing and addresses questions of fundamental importance and interest in mathematics. As such, it is a "pure research". However, it may have a considerable impact in the future in such areas as, e.g. High-Energy and Nuclear Physics. The current project is a natural continuation and development of the principal investigator's research program of last 15 years. His earlier results obtained in this program already find application in the recent work of several physicists (especially those associated with CERN -- European Laboratory of Particle Physics in Geneve).