Mathematical Sciences: Topological Index for Proper Actions, Asymptotic Homomorphisms and Equivariant E-Theory

抽象鳟鱼 所提出的研究涉及定义椭圆微分算子的等变“Atiyah-Singer”型拓扑指数，该指数对于局部紧群在基础流形上的真作用和余紧作用是不变的，即该指数 产生于等变嵌入基流形到一个（可能是无限维的）欧几里德表示群，并应用一个适当版本的等变博特周期。证明了相应的等变指数定理，即该拓扑指数等于G。Kasparov的椭圆算子的等变解析指标，需要使用C*-代数的等变渐近同态和A. Connes和N.希格森该研究对Baum-Connes猜想和 诺维科夫猜想 本文研究椭圆型微分算子的指数理论 在与局部紧变换群的作用可交换的流形上， 也就是说，研究具有一定内部对称性的微分方程的解的空间。与每个椭圆算子相关联的是一个“解析指数”， 其可以携带关于与算子相关联的流形的重要拓扑和几何信息。然而，这一分析指数一般很难计算，需要进行大量的详细分析。我建议构造，在一个更拓扑和几何的方式，另一种方法来计算这个指数，推广的情况下，紧凑的群体认为阿蒂亚和辛格。能够计算这些椭圆算子的指数应该会导致对几何和表示论中的各种问题有更清晰的理解，这些问题对物理学很重要。 特别是该项目将对鲍姆-康纳斯猜想和 尚未解决的诺维科夫猜想

Abstract Trout The proposed research involves defining an equivariant ``Atiyah-Singer''-type topological index for elliptic differential operators which are invariant with respect to the proper and cocompact action of a locally compact group on the underlying manifold, i.e. this index arises from equivariantly embedding the base manifold into a (possibly infinite dimensional) Euclidean representation of the group and applying an appropriate version of equivariant Bott periodicity. The proof of the corresponding equivariant index theorem, namely that this topological index is equal to G. Kasparov's equivariant analytic index for the elliptic operator, requires the use of equivariant asymptotic homomorphisms of C*-algebras and the associated equivariant version of the E-theory groups of A. Connes and N. Higson. This research has applications to the Baum-Connes Conjecture and the Novikov Conjecture. The proposed research deals with the index theory of elliptic differential operators on manifolds which commute with the action of a locally compact transformation group, i.e., studying the spaces of solutions of differential equations which exhibit certain degrees of internal symmetry. Associated with each elliptic operator is an "analytic index," which can carry important topological and geometric information about the manifold the operator is associated with. However, this analytic index is very difficult to compute in general, requiring a great deal of detailed analysis. I propose to construct, in a more topological and geometric fashion, another method for computing this index, generalizing the case of compact groups considered by Atiyah and Singer. Being able to compute the indices of these elliptic operators should lead to a clearer understanding of a variety of problems in geometry and representation theory, which are important to physics. In particular the project will have a bearing on the Baum-Connes Conjecture and the still-unsolved Novikov Conjecture.