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

摘要所提出的研究涉及为椭圆微分算子定义一个等变的‘Atiyah-Singer’型拓扑指数，该指数相对于局部紧群在基础流形上的真紧作用是不变的，即这个指数是通过将基流形等变地嵌入到群的欧几里得表示(可能是无穷维的)中并应用适当形式的等变Bott周期而产生的。要证明相应的等变指数定理，即这个拓扑指数等于G·Kasparov关于椭圆算子的等变解析指数，需要使用C*-代数的等变渐近同态以及A.Connes和N.Higson的E-理论群的相应的等变形式。这一研究适用于Baum-Connes猜想和Novikov猜想。本文研究在局部紧变换群作用下交换的流形上的椭圆微分算子的指标理论，即研究具有一定内部对称性的微分方程解的空间。与每个椭圆算子相关联的是一个“解析索引”，它可以携带关于该算子所关联的流形的重要的拓扑和几何信息。然而，这一分析指标总体上很难计算，需要进行大量的详细分析。我建议以更拓扑性和几何性的方式构造另一种计算这一指数的方法，推广了Atiyah和Singer所考虑的紧群的情况。能够计算这些椭圆算子的指数应该会使我们更清楚地理解几何和表示论中的各种问题，这些问题对物理很重要。特别是，该项目将对鲍姆-康尼斯猜想和仍然悬而未决的诺维科夫猜想产生影响。

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.