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

小行星9802493 库恩教授在发展同伦理论和表示论方法来解决这两个领域的问题方面有着悠久的历史。 最近，其中一些方法也导致他在代数K理论的方向，他的研究和技术正在广泛使用的其他人。 该项目的最大部分是针对连接经典的loopspace理论与最近的发展'S-模块'和'古德威利演算'技术，这些努力的重点是建立新的体系结构有关众所周知的和许多研究的拓扑对象：艾伦伯格-麦克莱恩空间，霍普夫不变量，和组合函数空间模型。 第二部分的工作是快速发展的一部分，现在在代数K理论和一般表示理论在有限领域的库恩教授，K理论家埃里克弗里德兰德和安德烈苏斯林，和他们的合作者。 最后，库恩教授和他的学生正在继续工作的拓扑实现问题，使用新的同伦理论和代数技术，以下方向的早期工作库恩教授和莱昂内尔施瓦茨的巴黎。 同伦理论、K理论和表示论都是试图发现并最终分类各种数学结构的基本“积木”的数学学科。 (This就像化学家研究简单的分子结构，以及这些分子如何以更复杂的方式组装。） 同伦涉及几何对象的变形，如高维表面。表示论涉及更刚性和离散对象的代数对称性，如线和平面的配置。 最后，K理论是两者的混合体。 库恩教授正在研究与这些主题相关的有趣的新联系，使用各种最先进的代数和同伦理论工具，其中许多是他自己开发的。

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.***