Homotopy Theory and its Applications

DMS-0203205Stephen A.Mitchell本文主要研究K(1)-局部同伦理论及其在代数K-理论中的应用。该项目的一个目标是计算数字环的一般线性群的mod2同调。一个更广泛的目标是从岩泽理论的角度系统地研究K(1)-局部同伦理论，以期应用于数字环的代数K-理论谱。现代数学最显著的特征之一是它的本质统一性。人们一次又一次地发现，这门学科看似互不相关的分支以意想不到的方式相互作用。代数拓扑学和数论的相互作用就是这种现象的一个值得注意的例子。数论最初是一个“离散”的学科，然后在19世纪转变为解析，并在20世纪变得越来越几何。在过去的三十多年里，代数拓扑学(研究几何对象，如曲线和曲面，在变形下保持不变的性质)越来越多地进入数论问题。这项拟议的工作将通过分析与经典数论数据相关联的某些几何对象的拓扑结构，进一步探索这两个学科之间的显著联系。

DMS-0203205Stephen A. MitchellThe proposed work is centered on K(1)-local homotopy theory and its applications to algebraic K-theory. One goal of the project is to compute the mod 2 homology of the general linear group of a number ring. A broader objective is a systematic study of K(1)-local homotopy theory from the perspective of Iwasawa theory, with a view toward applications to the algebraicK-theory spectra of number rings.One of the most striking features of modern mathematics is its essential unity. Time and again one finds that seemingly unrelated branches of the subject interact in unexpected ways. The interaction of of algebraic topology and number theory is a noteworthy example of this phenomenon. Number theory began as a "discrete" subject, then turned analytic in the 19th century and increasingly geometric in the 20th. In the last thirty years or so, algebraic topology (the study of propertiesof geometric objects, such as curves and surfaces, that remain invariant under deformation) has entered more and more into number-theoretic questions. The proposed work will explore further the remarkable connections between these two subjects, by analyzing the topology of certain geometric objects that can be associated to classical number-theoretic data.