Mathematical Sciences: Equivariant KK Theory

该奖项支持杰弗里·福克斯（Jeffrey Fox）的研究。 弗吉尼亚理工学院和州立大学的彼得·哈斯克尔 大学）的等变KK理论。 等变K理论和 交叉积代数K-理论在研究交叉积代数中 在解决几何和拓扑问题中的作用 与群作用和基本群有关 流形 这些问题的例子包括许多情况下， 关于度量存在性的Gromov-Lawson-Rosenberg猜想 具有正的标量曲率和许多诺维科夫 关于高阶签名同伦不变性的猜想。 Kasparov表示环KK（G）（C，C）及其区别 幂等元在这个程序中起了重要作用。 这 研究将应用卡斯帕罗夫表示理论与 目的是促进对地球的几何理解， 交叉积C（*）-代数的K-理论 这项“现代分析”的工作融合了复杂的研究 在代数、几何和分析中， 流形或曲面的理论。 这种分析是通过 空间变换群在曲面上的作用，或 在这个空间中的几何物体。

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.