Problems in Homotopy, K-theory, and Representation Theory

nicholas J. Kuhn教授长期致力于开发解决同伦理论、k理论和表示理论中有趣问题的方法。该项目最大的目标是将经典函数空间的新“多项式”分辨率与其他事物联系起来：结构谱的模型类别、等变稳定同伦、经典循环空间理论和周期同伦。这些连接将用于计算这些函数空间和无限循环空间的先前不可达的上同调不变量。第二部分的工作是有限域的一般表示理论研究的延续。特别是，从他的同调/ k理论的观点来看，他正在研究他新发现的广义舒尔代数的格，以期获得关于有限置换群和有限一般线性矩阵群的模表示的见解。同伦理论、k理论和表示理论是试图发现并最终分类各种数学结构的基本组成部分的数学学科。（这很类似于化学家研究简单的分子结构，以及如何以更复杂的方式组装这些结构。）同伦研究的是高维曲面等几何物体的变形。一个典型的难题是理解从球体（球的表面）到它自身的所有连续函数的“形状”。表征理论关注的是更刚性和离散对象的代数对称性，例如线和面的构型。一个示例难题是理解所有的基本方法，其中n × n的0和1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。库恩教授通过开发和使用各种最先进的代数和同伦理论工具，正在研究与这些学科相关的新联系。

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.