Finite group schemes and semi-topological theories

Friedlander提出的研究的一个方面是推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的基本例子是抽象代数的初学者遇到的第一个例子之一，但它的表示理论是“狂野的”，因此不可能列出所有有限维的表示。 弗里德兰德介绍了适用于这个和其他具体例子的建设性技术，但扩展到非常普遍的情况。 弗里德兰德建议继续他的研究表示任意有限群计划使用的见解和技术，从代数几何以及更传统的技术代数。 一个目标是有助于理解的具体例子;第二个目标是勾勒出一个一般的理论，其中包括这些例子;和第三个目标是利用某些特殊行动，研究代数K理论的某些奇异投影品种有限群计划。 第二个方面的弗里德兰德提出的研究是precision代数周期的代数品种。 这是代数几何中最基本和最具挑战性的课题之一，在过去的一百年里进行了大量的研究。 弗里德兰德的重点将是代数等价类的周期，影响的见解，从更好地理解模拟代数拓扑结构。应用设想代数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（有限群计划）反映在原来的问题，特别是在基本的，熟悉的例子？ 几何学如何，在第一次未被识别，约束的可能性，并导致具体的例子？ 这些例子的明确性质能在抽象的上下文中给出结构吗？这些是弗里德兰德建议与几位合作者一起调查的一些问题。 此外，他还建议使用代数拓扑学（形状理论）中开发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）在他的追求。 他还计划继续在出版数学，组织数学活动和服务于国家数学界的积极作用。

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.