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

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。本项目旨在计算接近霍普金斯-米勒高实k理论谱，结构环谱的代数k理论，和动力同伦。在这三种情况下，计算本身是紧密相关的：在有限群的代数作用（固定形式群的自同构，拓扑Hochschild同调上的圆作用，或伽罗瓦群的作用）和分类空间上的Thom谱的几何之间存在着微妙的相互作用。特别是，我们的目标有三个：(1)找到一个系统地确定Hopkins- miller高实k理论谱EO_n(G)的同伦环的程序（这是与Hopkins和Ravenel联合），(2)更好地支持Bokstedt-Hsiang-Madsen TR和TC机制以及基本色谱的代数k理论，(3)使用代数拓扑的标准技术提供动力同伦的基础计算。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。这些联系是自我强化的：代数中的问题变成了拓扑中的问题，而拓扑又被进一步细化为代数，而许多现代代数拓扑在很大程度上依赖于空间本身可以被更代数地描述的方式。该项目以多种方式利用了这种联系。首先，在试图理解如何用球体构建空间时，人们会遇到计算一大组不变量的问题：球体的同伦群。自20世纪30年代以来，这一直是代数拓扑学的一个非常活跃的部分，项目的第一部分是计算其他相关的环，它们作为越来越好的近似值。其次，最近的发展允许环和结构光谱的经典问题之间的双向交换。特别是，这个项目试图用更多的几何结构来更好地理解代数对象。

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.