Mathematical Sciences: Cyclic Homology, Higher Indices and Secondary Invariants

莫斯科维奇教授将利用非交换微分几何和非交换调和分析的工具来研究多重连通流形的微分和拓扑不变量。他的项目的第一部分涉及索引类型的不变量，是他在诺维科夫猜想上的工作的延续。第二部分，它代表了一个新的研究路线，处理更微妙的二次型不变量（例如解析扭转和eta不变量），并旨在将它们置于循环（co）同调和代数k理论的框架中。流形是光滑二维曲面的高维泛化。例如，时空连续体是一个四维流形。莫斯科维奇教授的研究旨在通过利用无限维空间中的线性变换（无限矩阵）理论的联系来理解这些几何对象。

Professor Moscovici will exploit the tools of non-commutative differential geometry and of non-commutative harmonic analysis for the investigation of differential and topological invariants of multiply connected manifolds. The first part of his project concerns invariants of index type and is a continuation of his work on the Novikov conjecture. The second part, which represents a new line of research, deals with the more subtle secondary-type invariants (e.g. analytic torsion and the eta invariant) and aims to place them in the framework of cyclic (co)homology and algebraic K-theory. A manifold is the generalization to higher dimensions of a smooth two dimensional surface. For example, the space-time continuum is a four dimension manifold. Professor Moscovici's research is aimed at understanding these geometric objects by exploiting connections with the theory of linear transformations (infinite matrices) in infinite dimensional spaces.