Number Theory, Combinatorics and Representation Theory (Mathematics)

这个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Deligne对广义Kloosterman和的估计和有限域上曲线的黎曼假设构造了Ramanujan图和具有小特征值的图。这种图有广泛的应用，所有已知的结构都是数论的。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，研究她构建的拉马努金图来自于通过算术离散子群连接到PGL2的树的商，并使用数论构建其他拉马努金图。第二部分是表征理论。它涉及使用用相关的y-因子表示的简化迹的公式证明局部四元数群表示的Tunnell的多重公式，并理解y-因子在局部Weil群的二级不可约表示上的作用。这些信息将增强对y因子如何决定表征类别的理解。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、表示理论和组合学方面发表几次研讨会演讲。该项目进一步推进了VPW计划的目标，即(1)为女性提供在工程和NSF支持的科学学科中发展职业的机会；(2)通过为在工业、政府和学术机构工作的女性科学家和工程师提供更大的知名度，鼓励女性在科学和工程领域追求职业。通过鼓励妇女参与科学，这是对国家未来科学活力的宝贵投资。

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.