Mathematical Sciences: Homotopy Theory and Its Applications

米切尔计划继续研究代数 K 理论的同伦论方面，重点关注与代数整数环的利希滕鲍姆-奎伦猜想相关的问题。 戈尔斯将继续他在不稳定同伦理论方面的工作。 特别是，他计划继续执行 Michael Barratt 的 Hopf 不变量分析计划，继续正在进行的 Hopf 代数研究，并研究 p-局部同伦理论的一些代数方面。 德维纳茨的研究继续了他对生成假说的研究和对莫拉瓦工作的阐述。 他还计划研究望远镜猜想的结果，特别是对悬浮光谱的布斯菲尔德类的结果。 这三个部分的细节各不相同，但都涉及将几何信息减少到计算主题或完善用于此目的的主要代数工具之一。 所涉及的几何信息的性质是困难的关键。 虽然关于长度、面积、角度、体积等的问题实际上迫切需要简化为计算，但它与所谓的几何对象的拓扑性质有很大不同。 这些属性包括连通性（全部为一体）、打结性、无孔等。 所有对这些性质的系统研究，例如，如何判断两个几何对象在这些性质之一上是否确实存在差异，或者只是表面上的差异，或者如何对可能出现的各种差异进行分类，所有这些只有在简化为计算问题时才真正被理解和掌握。 同伦理论和代数 K 理论已发展成为实现此目的的主要工具，并且代数和所涉及的拓扑之间的相互作用仍然是一个令人着迷的主题。

Mitchell plans to continue his investigation of homotopy- theoretic aspects of algebraic K-theory, focusing on questions related to the Lichtenbaum-Quillen conjectures for rings of algebraic integers. Goerss will continue his work on unstable homotopy theory. In particular, he plans to pursue Michael Barratt's program for analyzing Hopf invariants, to continue an ongoing study of Hopf algebras, and to study some of the algebraic aspects of p-local homotopy theory. Devinatz' research continues his study of the generating hypothesis and his exposition of Morava's work. He also plans to examine the consequences of the telescope conjecture, particularly to Bousfield classes of suspension spectra. The details of these three parts vary, but all are concerned either with reducing geometric information to a subject for calculation or to perfecting one of the principal algebraic tools used for this purpose. The nature of the geometric information involved is the crux of the difficulty. While questions about lengths, areas, angles, volumes, and so forth virtually cry out to be reduced to calculations, it is far different with what are known as topological properties of geometric objects. These are properties such as connectedness (being all in one piece), knottedness, having no holes, and so forth. All systematic study of such properties, for example, how to tell whether two geometric objects really differ in respect to one of these properties or are only superficially different, or how to classify the variety of differences that can occur, all these have only truly been comprehended and mastered when they have been reduced to matters of calculation. Homotopy theory and algebraic K-theory have been developed into major tools for this purpose, and the interplay between the algebra and the topology involved remains a fascinating subject.