Number Theory, Combinatorics and Representation Theory (Mathematics)

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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.