Mathematical Sciences: Presidential Young Investigator Award

霍普金斯最近一直在研究稳定同伦理论中幂零定理的结果。幂零定理决定了作为环到同伦的球谱的素理想。球体类似于整数环，是环的适当范畴中的初始对象。当一个人观察到球在其素理想处的“补全”获得自然紧致拓扑时，这个类比变得更加明显。他想把这些补全组合成一个全局对象，类似于数论中的阿德尔斯环。这样一个对象将适合于研究稳定同伦理论的大尺度方面。一个目标是构造具有解析性质的l序列，测量有限复的同伦群和它们之间的映射引起的同态。它的基础是推广k理论的上同理论族，一个基本问题是用“广义向量束”在几何上表示它们。Atiyah的一个定理将有限群的分类空间的k理论与其特征环联系起来。D. Ravenel和Hopkins最近概括了性格的概念，并为这些奇异的k理论证明了类似于Atiyah的结果。我们希望这种构造能够阐明“广义向量束”应该是什么。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，结性，稳定性等等。对于应用程序来说，这些特征与通过经典分析方法检查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.