Number Theory, Combinatorics and Representation Theory (Mathematics)

数论、组合学和表示论（数学）

• 批准号: 9003126
• 负责人: Wen-Ching Li
• 金额: --
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-06-15 至 1993-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9003126&HistoricalAwards=false
• 关键词: Number; Theory; Combinatorics; Representation; Mathematics

This project consists of two independent parts. The first part arises from Dr. Li's recent constructions of Ramanujan graphs and graphs with small eigenvalues using Deligne's estimate of generalized Kloosterman sums and using the Riemann hypothesis for curves over finite fields. Such graphs have wide applications and all the known constructions are number-theoretic. Continuing the theme of her work, she proposes to obtain estimates of interesting exponential sums from the Riemann hypothesis and Weil conjectures, to study which Ramanujan graphs she constructed come from the quotient of the tree attached to PGL2 by arithmetic discrete subgroups, and to construct other Ramanujan graphs using number theory. The second part is in representation theory. It concerns proving Tunnell's multiplicity formula for representations of a local quaternion group using the formula for the reduced traces expressed in terms of the associated y- factors and understanding the role of the y-factors attached to degree two irreducible representations of local Weil groups. The information will enhance the understanding of how the y-factors determine the classes of representations. In addition, Dr. Li will teach an advanced graduate course entitled "General Reciprocity Laws," and she will present several seminar talks in number theory, representation theory and combinatorics. This project furthers VPW program objectives which are (1) to provide opportunities for women to advance their careers in engineering and in the disciplines of science supported by NSF and (2) to encourage women to pursue careers in science and engineering by providing greater visibility for women scientists and engineers employed in industry, government, and academic institutions. By encouraging the participation of women in science, it is a valuable investment in the Nation's future scientific vitality.

该项目由两个独立的部分组成。 第一部分源于李博士最近使用德利涅对广义克洛斯特曼和的估计以及有限域上曲线的黎曼假设构建的拉马努金图和小特征值图。 此类图具有广泛的应用，并且所有已知的构造都是数论的。 继续她的工作主题，她建议从黎曼假设和韦尔猜想中获得有趣的指数和的估计，研究她构建的哪些拉马努金图来自通过算术离散子群附加到 PGL2 的树的商，并构建其他拉马努金使用数论绘制图表。 第二部分是表示论。 它涉及使用以相关 y 因子表示的约简迹公式来证明局部四元数群表示的 Tunnell 重数公式，并理解附加到局部 Weil 群的二次不可约表示的 y 因子的作用。 这些信息将增强对 y 因子如何确定表示类别的理解。 此外，李博士还将教授一门名为“一般互惠定律”的高级研究生课程，并将在数论、表示论和组合学方面发表多次研讨会演讲。 该项目进一步推进了 VPW 计划的目标，即 (1) 为女性提供在 NSF 支持的工程和科学学科领域推进职业生涯的机会，以及 (2) 通过提高女性在科学和工程领域的知名度，鼓励女性从事科学和工程领域的职业生涯。受雇于工业、政府和学术机构的女科学家和工程师。 通过鼓励女性参与科学，这是对国家未来科学活力的宝贵投资。