CAREER: p-adic harmonics, periods, and probability in lambda-rings

This project will build new connections between research frontiers in different fields of mathematics while providing innovative training opportunities for graduate and undergraduate students. The research areas developed will mix contributions by graduate students, undergraduates, and senior faculty by focusing on problems that admit entry points for researchers at different levels. The project will also develop specific new training programs, including an online bridge to advanced mathematics program for undergraduate freshmen and sophomores and a distinguished lecture series emphasizing interactions between the lecturer and local graduate students. The project focuses on research problems in three areas: p-adic harmonic analysis, periods of algebraic and analytic varieties, and probability in lambda rings. These areas connect to central problems in modern number theory, especially those arising from the study of the cohomology of algebraic and analytic varieties and the Langlands program. The project will develop new techniques that can be applied to important open problems in these fields such as the Fontaine-Mazur conjecture, which generalizes Wiles’ work on Fermat’s Last Theorem, and the Birch and Swinnerton-Dyer conjecture and its generalizations linking special values of L-functions to arithmetic. The tools that will be developed in the study of p-adic harmonic analysis and periods of algebraic and analytic varieties lie at the interface of geometry, analysis, and number theory, while the tools developed for the study of probability in lambda rings lie at the interface of probability theory, number theory, and commutative algebra. This project will thus elucidate new connections between these different areas of mathematics. Moreover, by focusing on problems that are accessible at different levels, and by developing targeted training opportunities for graduate and undergraduate students, the project will integrate the discovery of new mathematics with the development and mentoring of the next generation of US researchers in mathematics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.