Finite group schemes and semi-topological theories

有限群方案和半拓扑理论

• 批准号: 0909314
• 负责人: Eric Friedlander
• 金额: $ 17.94万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0909314&HistoricalAwards=false
• 关键词: Finite; group; schemes; semi; topological

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.

弗里德兰德（Friedlander）提出的研究的一个方面是促进对有限群体及其概括如何对向量空间作用的理解。 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，但其代表理论是``野性''，因此不可能列出所有有限的维度表示。弗里德兰德（Friedlander代数的几何形状以及更传统的代数技术是有助于对特定示例的理解；代数循环在代数品种上。 这是代数几何学的最基本和最具挑战性的主题之一，在过去的百年中进行了很多研究。 弗里德兰德（Friedlander）的重点将放在代数等效的周期类别上，这受到代数拓扑中更好理解的类似物的见解的影响。设想应用程序为代数K理论以及代数几何形状。 有限的对称组如何在有限场上或更一般的字段上对向量空间作出作用？ 对更一般的代数对象（有限组方案）的考虑如何反映出原始问题，尤其是在基本的，熟悉的示例中？ 几何形状最初是未被认可的，如何限制可能性并导致具体的例子？ 这些例子的明确性能可以在抽象环境中提供结构吗？这些是Friedlander提出的一些问题，要与几位合作者进行调查。 此外，他建议使用代数拓扑（形状理论）中开发的技术研究多项式方程（代数几何）的解决方案集。 弗里德兰德（Friedlander）计划鼓励年轻的数学家（包括他的过去，现在和未来的学生）的追求。 他还计划在出版数学，组织数学事件以及为国家数学社区服务方面继续他的积极角色。