Evaluation of L-series and Explorations in Mathematics

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9700542 Lieman This award funds programs for research and education centered around the areas of number theory and computational mathematics. In number theory, the investigator will address several outstanding conjectures about the average values of various families of L-series of arithmetic interest, including an "on average" version of the Lindelof hypothesis in n-aspect; a conjecture concerning the Whittaker-Fourier coefficients of metaplectic forms; and a conjecture concerning the average values of the L-series of the classical "sum of two cubes" family of elliptic curves. The investigator expects to resolve each conjecture by using and developing a blend of techniques from metaplectic forms and analytic number theory. In addition, the grant includes a summer research project in number theory for undergraduates. This summer program has diverse goals: to increase underrepresented minority participation in mathematics and science; to produce a self-paced mathematical investigation series which will be available to the general on-line undergraduate community; to involve undergraduates in active research investigations; and to produce new courses for the mathematics department at the University of Missouri. Finally, the investigator will undertake an experimental effort to increase graduate student visibility, travel opportunities, and professional experience within the mathematics community. The National Science Foundation strongly encourages the early development of academic faculty as both educators and researchers. The Faculty Early Career Development (CAREER) Program is a Foundation-wide program that provides for the support of junior faculty within the context of their overall career development. It combines in a single program the support of quality research and education in the broadest sense and the full participation of those traditionally underrepresented in science and engineering. This program enhances and emphasizes the importance the Foundation places on the development of full, balanced academic careers.