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n周期局域化理论。他将探索周期结果的不稳定版本，并将通过将它们分解为正整数n的n个周期部分来分析同伦类型。Bousfield正在建立v1周期局部化与空间的k理论局部化之间的紧密联系，这将导致有关迭代循环空间k理论的老问题的解决。他将继续他的工作，使用同调和k理论技术。例如，他将致力于发展k理论的不稳定亚当斯谱序列。Bousfield还将扩展他的k -局部谱的代数分类，以涵盖映射类，并将继续进行其他相关研究。这些部分的细节各不相同，但都与将几何信息简化为计算的主题或完善用于计算的代数机制有关。所涉及的几何信息的性质是难点的关键。虽然关于长度、面积、角度、体积等问题实际上迫切需要简化为计算，但这与几何对象的拓扑特性大不相同。这些属性包括连通性（整体）、打结性、无孔等等。所有对这些性质的系统研究，例如，如何判断两个几何物体在其中一种性质上是否真的不同，或者只是表面上的不同，或者如何对可能出现的各种差异进行分类，所有这些只有在它们被简化为计算问题时才真正被理解和掌握。其中两个主要的工具是同伦理论和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.