Mathematical Sciences: Presidential Young Investigator Award

霍普金斯最近一直在研究稳定同伦理论中的幂零定理的结果。幂零定理决定了被认为是同伦的环的球谱的素理想。球体类似于整数环，是合适的环类别中的初始对象。当人们观察到球体在其质数理想下的“完备性”获得自然紧致拓扑时，这种类比就变得更加惊人。他想把这些完成性组合成一个整体物体，类似于数论中的阿德莱斯环。这样的对象将适合于获得稳定同伦理论的大规模方面。一个目标是构造具有解析性质的L级数，该级数度量有限复形的同伦群以及它们之间的映射所诱导的同态。其基础是推广K-理论的一族上同调理论，一个基本问题是用“广义向量丛”来几何地表示它们。Atiyah的一个定理将有限群的分类空间的K-理论与它的特征标环联系起来。D.Ravenel和Hopkins最近推广了特征的概念，并证明了Atiyah关于这些奇异K-理论的结果的类比。人们希望，这一建设将阐明什么是“广义向量束”。所有这些活动都属于探索和完善处理空间定性特征的代数技术的总标题，例如连通性、结节性、稳定性等等。这些特征对于应用来说可能与经典分析方法所考察的通常特征一样重要。

Hopkins has recently been working out consequences of the nilpotence theorem in stable homotopy theory. The nilpotence theorem determines the prime ideals of the sphere spectrum thought of as a ring up to homotopy. The sphere is analogous to the ring of integers, being an initial object in a suitable category of rings. This analogy becomes more striking when one observes that the "completions" of the sphere at its prime ideals acquire natural compact topologies. He would like to assemble these completions into a global object analogous to the ring of adeles in number theory. Such an object would be appropriate for getting at large-scale aspects of stable homotopy theory. One goal would be to construct L-series with analytic properties measuring the homotopy groups of finite complexes and the homomorphisms induced by maps between them. Fundamental to this is a family of cohomology theories generalizing K-theory, and a basic problem is to represent them geometrically with "generalized vector bundles". A theorem of Atiyah relates the K-theory of the classifying space of a finite group to its character ring. D. Ravenel and Hopkins have recently generalized the notion of character and proved an analogue of Atiyah's result for these exotic K-theories. It is hoped that construction will shed light on what "generalized vector bundles" should be. All of this activity falls under the general heading of exploring and perfecting algebraic techniques for treating qualitative features of spaces, e.g. connectedness, knottedness, stability, and so forth. Such features can be every bit as important for applications as are the usual features examined by methods of classical analysis.