Matheamtical Sciences: Homotopy Theory and its Applications

9504476 Mitchell Mitchell， Goerss和Devinatz将研究同伦理论及其在代数k理论中的应用。目的是提高我们对这两门学科的基本问题的理解，包括场和整数环的k理论，同伦逆极限的性质，以及稳定同伦理论中的生成假设。(1) Mitchell将研究代数k理论的同伦理论方面，重点研究与代数整数环的Lichtenbaum-Quillen猜想有关的问题。特别是，他将进一步研究这种整数环的一般线性群的上同性。(2) Goerss研究的主要目的是解决一般情况下计算同伦逆极限的问题，以及在代数k理论中出现的群作用类型的同伦不动点的特殊情况下。其次是研究某些类型的Hopf代数，以及从空间的协链中恢复空间的同伦不变量的问题。(3) Devinatz将研究稳定同伦理论中的色观点。特别是，他对围绕霍普金斯的色度分裂猜想及其与生成假设的相关性的想法感兴趣。更基本的是，Mitchell、Goerss和Devinatz的项目都涉及到代数领域——尤其是数论——与更“几何”的同伦理论领域之间的关系，同伦理论可以被定义为研究在连续变形下保持不变的现象。米切尔计划用同伦理论来解释数论不变量，然后利用同伦理论的技术来获得对数论的新见解。仅使用代数技术可能难以获得这些见解。高斯将参与多个项目。其中一个涉及发展同伦理论中的技术，这些技术在米切尔的计划中应该是有用的。Goerss的另一个项目则反过来——也就是说，他将试图以完全代数的方式理解某些同伦不变量。Devinatz还将使用代数来深入了解同伦理论。在Devinatz的项目中，代数框架得到了更好的理解；然而，所涉及的代数相当复杂。因此，Devinatz必须尝试从代数中提取足够的信息以提供所需的同伦理论信息。这三位研究者也都有兴趣对围绕他们工作的思想圈提供更多的说明性描述；这些帐户应该使研究生更容易进入和掌握该领域。*＊*

9504476 Mitchell Mitchell, Goerss, and Devinatz will investigate homotopy theory and its applications to algebraic K-theory. The objective is to enhance our understanding of fundamental problems in both subjects, including the K-theory of fields and rings of integers, the nature of homotopy inverse limits, and the generating hypothesis in stable homotopy theory. (1) Mitchell will work on the homotopy theoretic aspects of algebraic K-theory, focusing on questions relating to the Lichtenbaum-Quillen conjectures for rings of algebraic integers. In particular, he will study further the cohomology of the general linear group of such rings of integers. (2) The principal aim of Goerss's research is to address the question of computing homotopy inverse limits in general and in the particular case of homotopy fixed points for the type of group action that arises in algebraic K-theory. Secondary to this main project is a study of certain types of Hopf algebras, and a study of the question of recovering homotopy invariants of a space from its cochains. (3) Devinatz will work on the chromatic point of view in stable homotopy theory. In particular, he is interested in the circle of ideas surrounding Hopkins's chromatic splitting conjecture and in its relevance to the generating hypothesis. More basically, the projects of Mitchell, Goerss, and Devinatz all involve the relationship between the field of algebra -- especially the theory of numbers -- and the more "geometric" field of homotopy theory, which may be defined as the study of phenomena that remain unchanged under continuous deformations. Mitchell plans to interpret number theoretic invariants in terms of homotopy theory and then to use the techniques of homotopy theory to gain new insights into number theory. These insights might be difficult to obtain using only algebraic techniques. Goerss will be working on a number of projects. One involves developing techniques in homotopy theory which should p rove useful in Mitchell's program. Another of Goerss's projects turns the flow around -- namely, he will attempt to understand certain homotopy invariants in a completely algebraic way. Devinatz will also use algebra to gain insights into homotopy theory. In Devinatz's project, the algebraic framework is better understood; however, the algebra involved is rather complicated. Devinatz must therefore try to extract enough information from the algebra to provide the desired homotopy theoretic information. All three investigators are also interested in providing more expository accounts of the circle of ideas surrounding their work; these accounts should make it easier for graduate students to enter and master the field. ***