Mathematical Sciences: Localization and Periodicity in Homotopy Theory

Bousfield将继续他在同伦理论的研究， 将特别关注新出现的VN周期理论 空间的局部化。 他将探索不稳定的 周期性结果，并将致力于分析同伦类型， 对于正整数n，将它们分解为vn-周期部分。 Bousfield正在建立v1周期性 局部化和空间的K-理论局部化， 是导致解决有关的K-理论的老问题 迭代循环空间。 他将继续他的工作，使用同伦 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发展。 Bousfield还将努力扩展他的代数分类， K-局部谱覆盖映射类，并将继续其他 相关研究。 这些部分的细节各不相同，但都涉及到 将几何信息简化为用于计算的对象，或者 完善用于计算的代数机器。 的 所涉及的几何信息的性质是关键的 困难 而长度、面积、角度、体积， 等几乎哭了，以减少计算，它是 与我们所知的 几何物体 这些属性比如连通性 （都在一块），打结，没有洞，等等 向前。 所有系统的研究，例如，如何 来判断两个几何对象是否真的在 这些属性之一，或者只是表面上的不同， 对可能发生的各种差异进行分类，所有这些 只有当他们真正理解和掌握， 减少到计算的问题，和两个主要的工具， 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.