Mathematical Sciences: Equivariant KK Theory

该奖项支持杰弗里·福克斯(与弗吉尼亚理工学院和州立大学的彼得·哈斯克尔联合)对等变KK理论的研究。等变K-理论和交叉积代数的K-理论在解决与群作用有关的几何和拓扑问题以及与流形的基本群有关的问题中发挥了重要作用。这类问题的例子包括许多关于具有正数量曲率的度量的存在的Gromov-Lawson-Rosenberg猜想和许多关于高签名的同伦不变性的Novikov猜想。Kasparov的表示环KK(G)(C，C)及其特殊的幂等元在本程序中起着基础性的作用。这项研究将应用Kasparov表示理论，以期有助于对交叉积C(*)-代数的K-理论的几何理解。《现代分析》的这项工作融合了代数、几何和分析的复杂研究，以考察流形或曲面的基本理论。这种分析是通过空间的变换群在该空间中的曲面或几何对象上的作用来进行的。

This award supports the research of Jeffrey Fox (joint with Peter Haskell of the Virginia Polytechnic Institute and State University) on equivariant KK-theory. Equivariant K-theory and the K-theory of crossed product algebras have played an important role in the solution of geometric and topological problems associated with group actions and with fundamental groups of manifolds. Examples of such problems include many cases of the Gromov-Lawson-Rosenberg conjecture on the existence of metrics with positive scalar curvature and many cases of the Novikov conjecture on the homotopy invariance of higher signatures. Kasparov's representation ring KK(G)(C,C) and its distinguished idempotent have played a fundamental role in this program. This research will apply the Kasparov representation theory with the intention for contributing to the geometric understanding of the K-theory of crossed product C(*)-algebras. This work in 'modern analysis' blends sophisticated research in algebra, geometry, and analysis to examine the underlying theory of manifolds or surfaces. This analysis is by way of the action of the transformation groups of a space on a surface or geometric object in that space.