Mathematical Sciences: Equivariant KK Theory

该奖项支持Peter Haskell（与科罗拉多大学的Jeffrey Fox合作）对等变kk理论的研究。等变k理论和交叉积代数的k理论在求解与群作用和流形基本群有关的几何和拓扑问题中发挥了重要作用。这类问题的例子包括许多关于正标量曲率度量存在性的Gromov-Lawson-Rosenberg猜想和许多关于高签名同伦不变性的Novikov猜想。卡斯帕罗夫的表示环KK(G)（C，C）及其杰出的幂等函数在该程序中发挥了重要作用。本研究将应用卡斯帕罗夫表示理论，旨在为交叉积C(*)-代数的k理论的几何理解做出贡献。这项工作在“现代分析”中融合了代数，几何和分析方面的复杂研究，以检查流形或曲面的潜在理论。这种分析是通过空间的变换群对该空间中的曲面或几何物体的作用来进行的。

This award supports the research of Peter Haskell (joint with Jeffrey Fox of the University of Colorado) 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.