Mathematical Sciences: Homotopy Theory and Its Applications

米切尔计划继续他对同伦的研究- 代数K理论的理论方面，侧重于问题 关于环的Lichtenbaum-Quillen定理， 代数整数 Goerss将继续他在Unstable的工作 同伦理论 特别是，他计划追求迈克尔 Barratt的分析Hopf不变量的程序，继续分析 正在进行的研究霍普夫代数，并研究一些代数 p-局部同伦理论 Devinatz的研究仍在继续 他对生成假说的研究和他对 摩拉瓦的作品 他还计划研究 望远镜猜想，特别是Bousfield类的 悬浮液光谱 这三个部分的细节各不相同，但都是有关的 或者通过减少对象的几何信息， 计算或完善一个主要的代数工具 用于此目的。 几何信息的本质 问题的症结在于。 虽然关于 长度、面积、角度、体积等等，几乎都在呼喊着， 如果我们把它简化为计算，它与我们所知道的大不相同。 几何物体的拓扑性质。 这些是 诸如连通性（所有的都是一体的）之类的属性， 结，没有洞，等等。 所有系统研究 例如，如何判断两个几何 对象在这些属性之一方面确实不同，或者 只是表面上的不同，或者如何分类的品种， 可能发生的差异，所有这些都只是真正的 当它们被简化为 计算. 同伦理论和代数K-理论一直是 发展成为主要的工具， 代数和拓扑之间的关系仍然是一个迷人的 话题吧

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.