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

Jean-Luc Brylinski研究了李群在流形上作用的上同不变量的计算和应用，以及李代数的无穷小作用。这些上同调不变量包括循环同调、等变上同调、等变k理论以及P. Baum、R. MacPherson和研究者的离域理论。中心的例子是光滑流形的自由环空间，在光滑流形上，圆的保向微分同态群D通过重参数化环来起作用。研究者将开发与循环同调和椭圆上同调有关的循环空间的微分几何工具。他将研究环空间上的d等变向量束，特别是那些与环空间表示相关的向量束，并将椭圆上同调与基于这些束的k理论联系起来。流形是研究的自然几何对象，通常作为常方程或微分方程的解集出现。作用于它们的变换群以一种对计算有用的形式对它们的对称性进行编码。（这在分析物理中出现的流形时经常被利用。）Brylinski所做的是开发新的上同调（代数）工具来研究流形上的变换群，然后将这些工具应用于重要的例子。

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.