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

这个奖项是根据2009年美国复苏和再投资法案(公法111-5)资助的。这个项目寻求从计算上接近Hopkins-Miller高实K-理论谱、结构环谱的代数K-理论和Motivic同伦。在这三种情况下，计算本身都是强相关的：有限群的代数作用(无论是固定形式群的自同构、拓扑Hochschild同调上的圆作用或Galois群的作用)与分类空间上的Thom谱的几何之间存在着微妙的相互作用。具体地说，我们的目标有三个：(1)找到一个程序来系统地确定Hopkins-Miller高次实K-理论谱EO_n(G)的同伦环(这是与Hopkins和Ravenel联合的)，(2)更好地理解Bokstedt-Hsiang-Madsen tr和TC机器以及基本色谱的代数K-理论，以及(3)使用标准的代数拓扑学技巧在模同伦中提供基础计算。代数拓扑学的目标是系统地在像数这样的代数对象和像空间这样的拓扑对象之间建立联系。这些联系是自我加强的：代数中的问题变成了拓扑学中的问题，而拓扑学又进一步细化为代数，而现代代数拓扑学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，在试图理解如何在球面之外构建空间时，人们遇到了计算一大族不变量的问题：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部分，该项目的第一部分是计算其他相关环，这些环充当越来越好的逼近。其次，最近的发展允许关于环的经典问题和结构谱之间的双向互换。特别是，这个项目寻求更好地理解代数对象，使用更多的几何结构。

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.