Finite group schemes and semi-topological theories

弗里德兰德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的初等例子是抽象代数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表示。弗里德兰德介绍了一些建设性的技术，这些技术适用于这个例子和其他具体的例子，但也适用于非常一般的情况。Friedlander建议使用代数几何的见解和技巧以及更传统的代数技巧来继续他对任意有限群方案的表示的研究。一个目标是有助于对具体例子的理解；第二个目标是勾画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与有限群方案相关的某些奇异射影簇的代数K-理论。Friedlander提出的研究的第二个方面是研究代数簇上的代数圈。这是代数几何中最基本和最具挑战性的课题之一，在过去的一百年里得到了大量的研究。弗里德兰德的重点将放在圈的代数等价类上，受代数拓扑学中更好理解的类比的影响。并展望了它在代数K理论和代数几何中的应用。有限对称群如何作用于有限域或更一般域上的向量空间？对更一般的代数对象(有限群方案)的考虑如何反映原始问题，特别是在基本的、熟悉的例子中？最初未被认识到的几何学如何限制可能性，并导致具体的例子？这些例子的显性性质能给出抽象上下文中的结构吗？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(形状论)中发展起来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)进行他的探索。他还计划继续在出版数学、组织数学活动和为全国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.