New Problems in Homotopy, K-Theory, and Representation Theory

库恩教授在发展同伦论和表示论方法来解决这两个领域的问题方面有着长期的记录。最近，这些方法中的一些方法也将他引向了代数K理论的方向，他的研究和技术正被其他人广泛使用。这个项目的最大部分是将经典的循环空间理论与最新的S-模和‘Goodwillie演算’技术的发展联系起来，这些努力的重点是建立关于著名和广泛研究的拓扑对象的新猜想：Eilenberg-MacLane空间、Hopf不变量和组合函数空间模型。这项工作的第二部分是由Kuhn教授、K-理论家Eric Friedlander和Andrei Suslin以及他们的合作者在有限域上的代数K-理论和一般表示理论方面取得的快速发展的一部分。最后，Kuhn教授和他的学生继续使用新的同伦理论和代数技术，沿着Kuhn教授和巴黎的Lionel Schwartz早期工作的方向，继续研究拓扑实现问题。同伦理论、K理论和表示理论是人们试图发现并最终归类各种数学结构的基本“积木”的数学学科。(这非常类似于化学家研究简单的分子构型，以及这些构型如何以更复杂的方式组装。)同伦研究的是几何对象的变形，如高维曲面。表象理论研究的是更多刚性和离散对象的代数对称性，如线和面的配置。最后，K-理论是两者的混合体。库恩教授正在使用各种最先进的代数和同伦理论工具，研究与这些学科相关的有趣的新联系，其中许多工具都是他自己开发的。

9802493Kuhn Professor Kuhn has a long record of developing both homotopy- theoretic and representation-theoretic methods to solve problems in both areas. More recently, some of these methods have led him also in algebraic K-theoretic directions, and his research and techniques are being used extensively by others. The largest part of this project is directed towards connecting classic loopspace theory with the more recent developments of `S-module' and `Goodwillie calculus' technology, with the focus of these efforts aimed towards establishing new conjectures concerning well known and much studied topological objects: Eilenberg-MacLane spaces, Hopf invariants, and combinatorial function space models. A second part of the work is part of the rapid development being made now in algebraic K-theory and generic representation theory over finite fields by Professor Kuhn, K-theorists Eric Friedlander and Andrei Suslin, and their collaborators. Finally, Professor Kuhn and his students are continuing work on topological realization problems, using both new homotopy theoretic and algebraic techniques, following the direction of earlier work by Professor Kuhn and Lionel Schwartz of Paris. Homotopy theory, K-theory, and representation theory aremathematical subjects in which one is trying to discover, andultimately classify, fundamental ``building blocks'' of various sortsof mathematical structure. (This is quite analogous to a chemist'sstudying simple molecular configurations, and how these can beassembled in more complex ways.) Homotopy is concerned withdeformations of geometric objects such as higher dimensional surfaces.Representation theory is concerned with the algebraic symmetries ofmore rigid and discrete objects such as configurations of lines andplanes. Finally K-theory is a hybrid of the two. Professor Kuhn isstudying intriguing new connections relating these subjects, using avariety of state-of-the-art algebraic and homotopy theoretic tools,many developed by himself.***