Problems in Homotopy, K-theory, and Representation Theory

DMS-0100710 Nicholas J. Kuhn Kuhn教授长期致力于开发解决同伦理论、K理论和表示论中有趣问题的方法。 这个项目的最大部分的目标是连接新的“多项式”决议的经典函数空间的其他事情：模型类别的结构谱，等变稳定同伦，经典循环空间理论和周期同伦。 这些连接将被用来计算以前无法访问这两个功能空间和无限循环空间的上同调不变量。 第二部分的工作是继续研究的通用表示理论的有限领域。 特别是，从他的同伦/K-理论的角度来看，他正在研究他新发现的广义舒尔代数格，着眼于获得有关有限置换群的模表示和有限的一般线性矩阵群的洞察力。同伦理论，K-理论和表示论是数学学科，其中一个试图发现，并最终分类，各种数学结构的基本组成部分。(This就像化学家研究简单的分子构型，以及如何以更复杂的方式组装这些分子。 同伦涉及几何对象的变形，如高维曲面。一个例子困难的问题是理解所有连续函数的“形状”从一个球（球的表面）到它本身。 表示论关注的是更刚性和离散对象的代数对称性，如线和平面的配置。一个困难的问题是要理解所有的基本方式，其中0和1的n乘n矩阵的集合可以作用于0和1的字符串。 最后，K理论是两者的复杂混合体。 库恩教授正在研究与这些主题相关的新联系，通过开发和使用各种最先进的代数和同伦理论工具。

DMS-0100710Nicholas J. KuhnProfessor Kuhn has a long record of developing methods to solve interesting problems in homotopy theory, K-theory, and representation theory. The goal of the largest part of this project is to connect new 'polynomial' resolutions of classic function spaces to other things: model categories of structured spectra, equivariant stable homotopy, classial loop space theory, and periodic homotopy. These connections will be used to calculate previously inaccessible cohomological invariants of both such function spaces and infinite loop spaces. A second part of the work is a continuation of the study of the generic representation theory of finite fields. In particular, from his homotopical/K-theoretic point of view, he is studying his newly discovered lattices of generalized Schur algebras, with an eye towards gaining insight about the modular representations of the finite permutation groups, and the finite general linear matrix groups.Homotopy theory, K-theory, and representation theory are mathematical subjects in which one is trying to discover, and ultimately classify, fundamental building blocks of various sorts of mathematical structure. (This is quite analogous to a chemist studying simple molecular configurations, and how these can be assembled in more complex ways.) Homotopy is concerned with deformations of geometric objects such as higher dimensional surfaces. A sample hard problem is to understand the `shape' of all continuous functions from a sphere (surface of a ball) to itself. Representation theory is concerned with the algebraic symmetries of more rigid and discrete objects such as configurations of lines and planes. A sample hard problem is to understand all the basic ways in which the set of n by n matrices of 0's and 1's can act on strings of 0's and 1's. Finally K-theory is a sophisticated hybrid of the two. Professor Kuhn is studying new connections relating these subjects, by developing and using a variety of state-of-the-art algebraic and homotopy theoretic tools.