Mathematical Sciences: Noncommutative Geometry

在这个项目中，Wodzicki将研究k理论和算子理想的循环同调，以及理想中带系数的Hilbert空间上有界算子代数的Hochschild同调。特别是，他将研究：行列式，调节器，高解析和拓扑指标不变量，Hilbert空间上有界线性算子的交换子结构，以及与代数变体k理论的联系。这种主题的结合自然是主要研究者先前工作的结果。该方法结合了几个数学领域，特别关注解析和几何意义的（非交换）代数的同调研究，以及合适的同调和同调工具的发展。该项目触及了现代数学发展的基本方向，并解决了数学的基本重要性和兴趣问题。因此，这是一项“纯粹的研究”。但是，它将来可能在诸如高能和核物理等领域产生相当大的影响。本课题是本课题负责人过去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（特别是那些与日内瓦的欧洲粒子物理实验室有关的物理学家）最近的工作中得到了应用。

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).