Maass forms and number theory

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Building on recent work on mock theta functions and mock modular forms, the PI proposes an ambitious program to investigate the interplay between mock modular forms and the theory of L-functions, elliptic curves, and representation theory. The PI will also aim to generalize the theory of harmonic Maass forms to seek explicit constructions of Maass cusp forms.The PI has had success in the related problem of explicitly computing harmonic forms. New results in these areas should have implications for the Birch and Swinnerton-Dyer Conjecture, the arithmetic of critical values of L-functions, Donaldson invariants, and integer partitions.This research in number theory has the potential for far reaching real world applications. Number theory plays a central role in cryptography, internet security, and the infrastructure of emerging electronic markets. Further theoretical advances in the proposed research will also have implications in theoretic physics (string theory), and probability theory (e.g. tumor modelling via special cellular automata).