Maass forms and number theory

基于最近对模拟theta函数和模拟模块化形式的研究，PI提出了一个雄心勃勃的计划，以研究模拟模块化形式与L函数理论，椭圆曲线和表示理论之间的相互作用。PI还将推广调和Maass形式的理论，以寻求Maass尖点形式的显式构造。PI在显式计算调和形式的相关问题上取得了成功。这些领域的新成果对Birch和Swinnerton-Dyer猜想、L-函数临界值的计算、唐纳森不变量和整数划分等都有重要意义，对真实的世界有着深远的应用前景。数论在密码学、互联网安全和新兴电子市场的基础设施中发挥着核心作用。在拟议的研究中，进一步的理论进展也将对理论物理（弦论）和概率论（例如通过特殊的细胞自动机进行肿瘤建模）产生影响。

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).