Mathematical Sciences: Localization and Periodicity in Homotopy Theory

Bousfield将继续他在同伦理论方面的研究，并将特别关注新出现的空间的vn周期局部化理论。他将探索周期性结果的不稳定版本，并将致力于通过将同伦类型分解为正整数n的vn-周期部分来分析同伦类型。Bousfield在空间的v1-周期局部化和K-理论局部化之间建立了强大的联系，这导致了关于迭代循环空间的K-理论的旧问题的解决。他将继续他的工作，使用同伦和K理论技术。例如，他将致力于K理论不稳定亚当斯谱序列的开发。Bousfield还将致力于扩展他对K-局部谱的代数分类，以涵盖映射类，并将继续其他相关研究。这些部分的细节各不相同，但都涉及将几何信息归结为用于计算的主题或完善用于计算的代数机器。涉及的几何信息的性质是困难的症结所在。虽然关于长度、面积、角度、体积等的问题实际上需要归结为计算，但它与几何对象的拓扑属性有很大的不同。这些属性包括连通性(完好无损)、多节、无洞等。所有这些性质的系统研究，例如，如何区分两个几何对象在这些性质中的一个是否真的不同，或者仅仅是表面上的不同，或者如何对可能发生的各种不同进行分类，所有这些都只有在归结为计算问题时才被真正理解和掌握，而这方面的两个主要工具是同伦理论和K理论。

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.