Computations in Classical Chromatic Homotopy Theory, Algebraic K-Theory, and Motivic Homotopy

经典色同伦理论、代数 K 理论和基元同伦的计算

• 批准号: 0906285
• 负责人: Michael Hill
• 金额: $ 10.09万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-15 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906285&HistoricalAwards=false
• 关键词: Computations; Classical; Chromatic; Homotopy; Theory

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project seeks to computationally approach the Hopkins-Miller higher real K-theory spectra, the algebraic K-theory of structured ring spectra, and motivic homotopy. In all three cases, the computations themselves are strongly related: there is a subtle interplay between the algebraic action of a finite group (be they automorphisms of a fixed formal group, the circle action on topological Hochschild homology, or the action of a Galois group) and the geometry of Thom spectra over the classifying spaces. In particular, our goals are threefold: (1) find a procedure to systematically determine the homotopy ring of the Hopkins-Miller higher real K-theory spectra EO_n(G) (this is joint with Hopkins and Ravenel), (2) better standing the Bokstedt-Hsiang-Madsen TR and TC machinery and the algebraic K-theory of basic chromatic spectra, and (3) use the standard techniques of algebraic topology to provide foundational computations in motivic homotopy.The goal of algebraic topology is to systematically build a connection between algebraic objects like numbers and topological objects like spaces. These connections are self-reinforcing: problems in algebra become problems in topology which are further refined into algebra, and much of modern algebraic topology relies heavily on the ways spaces themselves can be described more algebraically. This project exploits this connection in multiple ways. First, in trying to understand how to build spaces out of spheres, one encounters the problem of computing a large family of invariants: the homotopy groups of spheres. This has been a very active part of algebraic topology since the 1930s, and the first part of the project is to compute other, related rings which act as increasingly good approximations. Second, the recent developments allow a two-way interchange between classical questions about rings and structured spectra. In particular, this project seeks to better understand the algebraic objects using more geometrical constructions.

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该项目旨在计算方法的霍普金斯-米勒更高的真实的K-理论光谱，代数K-理论的结构环光谱，和motivic同伦。在所有这三种情况下，计算本身都是密切相关的：有限群的代数作用（无论它们是固定形式群的自同构，拓扑Hochschild同调的循环作用，还是伽罗瓦群的作用）与分类空间上的Thom谱的几何之间存在微妙的相互作用。具体而言，我们的目标有三个：（1）系统地确定Hopkins-Miller高真实的K理论谱EO_n（G）的同伦环（2）较好地代表了Bokstedt-Hsiang-Madsen的TR和TC机制和基本色谱的代数K理论，（3）使用代数拓扑的标准技术来提供运动同伦的基础计算。代数拓扑的目标是系统地建立代数对象（如数）和拓扑对象（如空间）之间的联系。这些联系是自我加强的：代数中的问题变成了拓扑中的问题，这些问题又被进一步细化到代数中，而现代代数拓扑在很大程度上依赖于空间本身可以更代数地描述的方式。这个项目以多种方式利用了这种联系。首先，在试图理解如何从球面构建空间时，人们遇到了计算一大类不变量的问题：球面的同伦群。自20世纪30年代以来，这一直是代数拓扑学中非常活跃的一部分，该项目的第一部分是计算其他相关的环，这些环作为越来越好的近似。其次，最近的发展允许一个双向交换的经典问题之间的环和结构谱。特别是，这个项目旨在更好地理解代数对象使用更多的几何结构。