Number Theory, Combinatorics and Representation Theory (Mathematics)

该项目由两个独立的部分组成。 第一部分 来自李博士最近的Ramanujan图的构造， 用Deligne的广义特征值估计 Kloosterman求和和使用黎曼假设的曲线超过 有限域 这样的图有广泛的应用，所有已知的 结构是数论的。 继续她工作的主题， 她建议从以下公式中获得有趣的指数和的估计： 黎曼假设和魏尔定理，研究拉马努金 她构造的图来自于树的商， PGL2通过算术离散子群，并构造其他 使用数论的Ramanujan图。 第二部分是在 表征理论 它涉及证明Tunnell的多重性 局部四元数群的表示公式， 以相关的y表示的简化迹线的公式 因素和理解Y因素的作用 局部Weil群的两个不可约表示 的信息 将增强对y因子如何决定 代表的类。 此外，李博士将教授一门高级研究生课程， “一般互惠法”，她将提出几个研讨会的讲话 在数论，表示论和组合数学中。 本项目进一步促进了VPW计划的目标，即（1） 为妇女提供机会， 工程和科学的学科由NSF支持， (2)鼓励妇女从事科学和工程职业， 提高女科学家和女工程师的知名度 受雇于工业、政府和学术机构。 通过 鼓励妇女参与科学，这是一个宝贵的 投资于国家未来的科学活力。

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.