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. ***
