Homotopy Theory and its Applications

同伦理论及其应用

• 批准号: 0203205
• 负责人: Stephen Mitchell
• 金额: $ 19.25万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-15 至 2006-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0203205&HistoricalAwards=false
• 关键词: Homotopy; Theory; its; Applications

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.

DMS-0203205 Stephen A.米切尔所提出的工作集中在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环的一般线性群的模2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，看似无关的学科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是这种现象的一个值得注意的例子。数论开始时是一个“离散”的学科，然后在世纪转向分析，在20世纪逐渐转向几何。在过去的30年左右，代数拓扑（研究几何对象，如曲线和曲面，在变形下保持不变的性质）已经越来越多地进入数论问题。拟议的工作将进一步探索这两个主题之间的显着联系，通过分析拓扑结构的某些几何对象，可以与经典数论数据。