Mathematical Sciences: Homotopy Theory and Its Applications

米切尔教授计划研究同伦理论的最新发展对代数 K 理论的影响。 特别是，他计划利用 Devinatz、Hopkins 和 Smith 的出色工作，从同伦理论的更高周期性的角度研究与 Lichtenbaum-Quillen 猜想相关的代数 K 理论的各个方面；重点是，根据托马森的工作，代数 K 理论中不应该有任何更高的周期性。 在一个相关项目中，他计划研究有限复形的周期自映射、几何实现 Steenrod 代数子代数的问题，以及从对称群的模表示论获得的稳定分裂。 这些研究将有助于整合研究几何空间的两种主要代数机器，即近年来发展特别迅速的同伦理论和当前热门探索领域的代数K理论。

Professor Mitchell plans to investigate the implications for algebraic K-theory of recent developments in homotopy theory. In particular, he plans to study aspects of algebraic K-theory connected with the Lichtenbaum-Quillen conjectures from the point of view of higher periodicity in homotopy theory, using the spectacular work of Devinatz, Hopkins and Smith; the point being that by work of Thomason there shouldn't be any higher periodicity in algebraic K-theory. In a related project he plans to study periodic self-maps of finite complexes, the problem of realizing subalgebras of the Steenrod algebra geometrically, and stable splittings obtained from the modular representation theory of symmetric groups. These investigations will help to integrate two of the major algebraic machines for investigating geometric spaces, namely homotopy theory, which has been advancing particularly rapidly in recent years, and algebraic K-theory, also the scene of intense current exploration.