Matheamtical Sciences: Homotopy Theory and its Applications

小行星9504476 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个学科的基本问题的理解，包括域和整数环的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)米切尔将致力于同伦理论方面的代数K-理论，侧重于有关问题的Lichtenbaum-Quillen代数整数环的结构。 特别是，他将进一步研究上同调的一般线性群的这种环的整数。 (2)Goerss研究的主要目的是解决计算同伦逆极限的问题，在一般情况下，在特定情况下同伦不动点的类型组行动中出现的代数K-理论。 次要的这个主要项目是研究某些类型的霍普夫代数，并研究问题的恢复同伦不变量的空间从其上链。 (3)Devinatz将致力于稳定同伦理论中的色观点。 特别是，他感兴趣的是圆的想法周围霍普金斯的色分裂猜想，并在其相关性的生成假说。 更基本的是，米切尔、戈斯和德文纳茨的项目都涉及代数领域（特别是数论）与同伦理论的更“几何”领域之间的关系，同伦理论可以定义为研究在连续变形下保持不变的现象。 米切尔计划解释数论不变量的同伦理论，然后使用同伦理论的技术，以获得新的见解数论。 这些见解可能很难获得仅使用代数技术。 Goerss将参与多个项目。 一个涉及同伦理论的发展技术，这应该在米切尔的程序中很有用。 Goerss的另一个项目将流程转过来-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。 这三位研究人员也都有兴趣提供更多关于他们工作周围思想圈的简要说明;这些说明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. ***