Harmonic Maass Forms, "Moonshine," and Arithmetic Statistics

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Einstein's general relativity was formulated just over a hundred years ago. This theory now underpins the Global Positioning System, an almost ubiquitous feature of modern life. Almost concurrently, the mathematical genius Ramanujan arrived in England to work with Cambridge mathematician G. H. Hardy. The observations and insights he shared then continue to fascinate the mathematical world, and his personal story has captured the public imagination. In recent years evidence has emerged that the work of Einstein and the work of Ramanujan are related. This research project aims to develop the theory underlying this connection and lay important groundwork for future applications to mathematics, signal processing, and physics. The PI will continue to train graduate students in mathematics as part of this award.This project is dedicated to arithmetic statistics and the development of the theory of harmonic Maass forms. These tools will be applied to various open problems in number theory, representation theory, and mathematical physics. Harmonic Maass forms generalize modular forms, and are now recognized as furnishing a theoretical framework for Ramanujan's mock theta functions, as well as the currently-developing umbral moonshine theory. Building on the PI’s earlier work in these areas, the goal of this project is to develop algebraic, analytic, and combinatorial tools for harmonic Maass forms that will shed light upon statistical problems such as ranks of elliptic curves, torsion in class groups of number fields, distribution of Fourier coefficients of modular forms, the relationship between monstrous and umbral moonshine, and the consequences of this relationship for physics.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.