Bousfield will continue his research in homotopy theory and will focus particularly on the newly emerging theory of vn-periodic localizations of spaces. He will explore unstable versions of periodicity results and will work to analyze homotopy types by resolving them into vn-periodic parts for positive integers n. Bousfield is establishing strong links between the v1-periodic localization and the K-theoretic localization of spaces, and this is leading to solutions of old problems concerning the K-theory of iterated loop spaces. He will continue his work, using homotopical and K-theoretic techniques. For instance, he will work on the development of a K-theoretic unstable Adams spectral sequence. Bousfield will also work to extend his algebraic classification of K-local spectra to cover mapping classes and will continue other related research. The details of these parts vary, but all are concerned with reducing geometric information to a subject for calculation or to perfecting the algebraic machinery used for the calculations. 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, and two of the principal tools for this are homotopy theory and K-theory.
