Mitchell plans to continue his investigation of homotopy- theoretic aspects of algebraic K-theory, focusing on questions related to the Lichtenbaum-Quillen conjectures for rings of algebraic integers. Goerss will continue his work on unstable homotopy theory. In particular, he plans to pursue Michael Barratt's program for analyzing Hopf invariants, to continue an ongoing study of Hopf algebras, and to study some of the algebraic aspects of p-local homotopy theory. Devinatz' research continues his study of the generating hypothesis and his exposition of Morava's work. He also plans to examine the consequences of the telescope conjecture, particularly to Bousfield classes of suspension spectra. The details of these three parts vary, but all are concerned either with reducing geometric information to a subject for calculation or to perfecting one of the principal algebraic tools used for this purpose. The nature of the geometric information involved is the crux of the difficulty. While questions about lengths, areas, angles, volumes, and so forth virtually cry out to be reduced to calculations, it is far different with what are known as topological properties of geometric objects. These are properties such as connectedness (being all in one piece), knottedness, having no holes, and so forth. All systematic study of such properties, for example, how to tell whether two geometric objects really differ in respect to one of these properties or are only superficially different, or how to classify the variety of differences that can occur, all these have only truly been comprehended and mastered when they have been reduced to matters of calculation. Homotopy theory and algebraic K-theory have been developed into major tools for this purpose, and the interplay between the algebra and the topology involved remains a fascinating subject.
