Mathematical Sciences: Function Spaces, Homotopy Theory, and Group Cohomology

在下面描述的项目中，有两位教师调查员，F. R. Cohen和S. Gitler，以及两位参与论文研究的研究生。所有的研究都涉及拓扑学中的代数技术。F. R. Cohen打算研究经典同伦理论中的一些问题以及某些离散群的上同伦的一些分支。三个主要项目包括：(1)改进他在模2的r次摩尔空间（r至少为2）的指数上的界，并尝试将一些结果推广到r为1的情况下，(2)研究包含有理函数空间的Whitehead积及其与Cayley-Dickson代数的相互作用，(3)一类称为映射类群的离散群的分类空间的上同调和稳定结构。其中一些问题涉及构型空间，是S. Gitler和L. Taylor（圣母大学）共同工作的一部分。S. Gitler打算考虑几个问题，集中在映射空间，它们的上同调，以及它们与同伦理论的关系。例如，他打算研究“好”空间X和Y的映射（X，Y）及其上同调。设Y等于Morava K理论的-谱空间K(n)，可以得到具有周期同伦群的空间的有用例子。吉特勒打算研究这些空间的应用。

There are two faculty investigators in the projects described below, F. R. Cohen and S. Gitler, as well as two of their graduate students involved in thesis research. All of the research concerns algebraic techniques in topology. F. R. Cohen intends to consider problems in classical homotopy theory and some offshoots to the cohomology of certain discrete groups. Three of the main projects involve (1) improving his bounds on the exponent of a mod-2-to-the-r Moore space, r at least 2, and trying to extend some results to the case when r is 1, (2) a study of the Whitehead product involving spaces of rational functions together with their interplay with Cayley-Dickson algebras, and (3) the cohomology and stable structure for the classifying space of certain discrete groups called mapping class groups. Some of these problems involve configuration spaces and are part of joint work with S. Gitler and L. Taylor (University of Notre Dame). S. Gitler intends to consider several problems centered on mapping spaces, their cohomology, and their relationship to homotopy theory. For example, he intends to study maps (X,Y) and their cohomology for "good" spaces X and Y. Setting Y equal to a space in the omega-spectrum for Morava K-theory, K(n), one gets useful examples of spaces with periodic homotopy groups. Gitler intends to study applications of these spaces.