Finite group schemes and semi-topological theories

弗里德兰德提出的研究的一个方面是促进对有限群及其推广如何作用于向量空间的理解。Z/p x Z/p的基本例子是初学抽象代数的学生最先遇到的例子之一，但它的表示理论是“疯狂的”，因此不可能列出所有有限维表示。弗里德兰德介绍了建设性的技巧，适用于这个和其他具体的例子，但扩展到非常普遍的情况。Friedlander建议使用代数几何的见解和技术以及更传统的代数技术继续研究任意有限群方案的表示。一个目标是促进对具体例子的理解；第二个目标是勾勒出一个包含这些例子的一般理论；第三个目标是利用某些特殊作用来研究有限群方案下某些奇异射影变的代数k理论。弗里德兰德提出的研究的第二个方面是对代数变量上的代数循环的研究。这是代数几何中最基本和最具挑战性的课题之一，在过去的一百年里，人们对它进行了大量的研究。Friedlander的重点将放在环的代数等价类上，受到代数拓扑中更好理解的模拟的影响。应用设想到代数k理论以及代数几何。有限对称群如何作用于有限域或更一般域上的向量空间？如何考虑更一般的代数对象（有限群方案）反映原来的问题，特别是在基本的，熟悉的例子？几何图形是如何在一开始被忽视的情况下限制可能性并导致具体的例子？这些例子的明确性质能否在抽象语境中给出结构？这些都是弗里德兰德提议与几位合作者一起研究的一些问题。此外，他建议使用代数拓扑（形状理论）中发展的技术来研究多项式方程（代数几何）的解集。弗里德兰德计划鼓励年轻的数学家（包括他过去、现在和未来的学生）参与他的探索。他还计划继续在出版数学、组织数学活动和为国家数学界服务方面发挥积极作用。

One aspect of Friedlander's proposed research is to advance the understanding of how finite groups and their generalizations act on vector spaces. The elementary example of Z/p x Z/p is one of the first examples encountered by beginning students of abstract algebra, yet its representation theory is ``wild" so that no listing of all finite dimensional representations is possible. Friedlander has introduced constructive techniques which apply to this and other specific examples yet extend to very general situations. Friedlander proposes to continue his study of representations of arbitrary finite group schemes using insights and techniques from algebraic geometry as well as more traditional techniques of algebra. One goal is to contribute to the understanding of specific examples; a second goal is to sketch a general theory which incorporates these examples; and a third goal is to utilize certain special actions to study the algebraic K-theory of certain singular projective varieties associated to finite group schemes. A second aspect of Friedlander's proposed research is the investigtion of algebraic cycles on algebraic varieties. This is one of the most fundamental and challenging topics of algebraic geometry, much studied in the past hundred years. Friedlander's focus will be on algebraic equivalence classes of cycles, influenced by insights from the better understood analogue in algebraic topology. Applications are envisioned to algebraic K-theory as well as algebraic geometry. How can finite groups of symmetries act on vector spaces over finite fields or over even more general fields? How does the consideration of more general algebraic objects (finite group schemes) reflect on the original problem, especially in basic, familiar examples? How does the geometry, at first unrecognized, constrain the possibilities and lead to concrete examples? Can the explicit nature of these examples give structures in abstract contexts? These are some of the questions Friedlander proposes to investigate with several collaborators. In addition, he proposes to study solution sets of polynomial equations (algebraic geometry) using techniques developed in algebraic topology (theory of shapes). Friedlander plans to encourage younger mathematicians (including his past, present, and future students) in his quest. He also plans to continue his active roles in publishing mathematics, organizing mathematical events, and serving the national mathematical community.