Mathematical Sciences: Group Actions on Manifolds, Cyclic Homology and Elliptic Cohomology

这个研究项目由让-吕克Brylinski处理 上同调不变量的计算及其应用 李群在流形上的作用，以及 李代数 这些上同调不变量包括循环的 同调，等变上同调，等变K-理论， Baum，R.麦克弗森和 调查员 中心示例将是 一个光滑流形，其上的保向群D 圆的反同态通过重新参数化循环而起作用。 的 研究人员将开发用于回路微分几何工具 空间的循环同调和椭圆上同调。 他将研究D-等变向量丛循环空间，在 特别是与循环空间的表示相关联的那些， 并将椭圆上同调与基于这样的 捆起来。 流形是研究的自然几何对象，通常 作为常微分方程的解集而产生。 作用于它们的变换群将它们的对称性编码为 这对计算很有用。 (This经常被利用 在分析物理学中出现的流形时。） 什么布林斯基 正在开发新的上同调（代数）工具， 研究作用在流形上的变换群， 将这些工具应用到重要的例子中。

This reserch project by Jean-Luc Brylinski deals with the computation and the applications of cohomological invariants for actions of Lie groups on manifolds, and infinitesimal actions of Lie algebras. These cohomological invariants include cyclic homology, equivariant cohomology, equivariant K-theory, and the delocalized theory of P. Baum, R. MacPherson and the investigator. The central example will be the free loop space of a smooth manifold, on which the group D of orientation-preserving diffeomorphisms of the circle acts by reparametrizing loops. The investigator will develop differential-geometric tools for loop spaces, in relation with cyclic homology and elliptic cohomology. He will study D-equivariant vector bundles on loop spaces, in particular those associated with representations of loop spaces, and relate elliptic cohomology to a K-theory based on such bundles. Manifolds are natural geometric objects to study, often arising as solution sets of ordinary or differential equations. Transformation groups acting upon them encode their symmetry in a form that is useful for computations. (This is often exploited in analysing manifolds which arise in physics.) What Brylinski is doing is developing new cohomological (algebraic) tools for studying transformation groups acting on manifolds and then applying these tools to important examples.