Applications of automorphic forms and hypergeometric q-series

自守形式和超几何 q 级数的应用

• 批准号: 1201435
• 负责人: Karl Mahlburg
• 金额: $ 13.4万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201435&HistoricalAwards=false
• 关键词: Applications; automorphic; forms; hypergeometric; series

The primary aim of this research program is to study the applications of modular and automorphic forms, and hypergeometric q-series. These applications include a wide selection of different areas of mathematics and mathematical physics, such as the theory of integer partitions, combinatorial probability and Markov processes, bootstrap percolation models, affine Lie superalgebras, and (quadratic) Hurwitz class numbers. The automorphic objects of interest include modular and Jacobi forms, as well as mock modular and Jacobi forms, which have seen a great deal of recent interest thanks to work of Borcherds, Bringmann, Bruinier, Funke, Ono, Zagier, and Zwegers. Mock modular forms are of particular interest due both to their connections with harmonic Maass forms, as well as their famous history as objects of mystery dating back to Ramanujan and Watson. As an example of the interplay between automorphic forms and other topics, the PI and Bringmann recently used combinatorial probability bounds for gap-avoiding sequences (that first arose in Holroyd, Liggett, and Romik's study of finite-size scaling in bootstrap percolation) in order to prove a cuspidal asymptotic expansion for a family of hypergeometric q-series considered by Andrews.Due to its scope, this research program has the potential for wide-ranging applications. An underlying theme is the universality of the tools and techniques of modern number theory, whose best-known uses include cryptography and cellular communication. This research will also illustrate applications to high-energy physics (where wall-crossings and black holes are described by mock theta functions), and biological 'growth' processes (where the the large-scale behavior of cellular automata is determined by the asymptotic expansions of modular forms).

这个研究项目的主要目的是研究模形式、自同构形和超几何Q-级数的应用。这些应用包括不同的数学和数学物理领域的广泛选择，例如整数划分理论、组合概率和马尔可夫过程、自举渗流模型、仿射李超代数和(二次)Hurwitz类数。自同构的感兴趣对象包括模块和Jacobi形式，以及模拟模块和Jacobi形式，由于BorCherds、Bringmann、Bruinier、Funke、Ono、Zagier和Zwegers的工作，这些形式最近引起了极大的兴趣。模拟模块形式特别令人感兴趣，因为它们与调和的Maass形式有联系，以及它们作为神秘对象的著名历史可以追溯到Ramanujan和Watson。作为自同构形式和其他主题相互作用的一个例子，PI和Bringmann最近使用了间隙回避序列的组合概率界(最早出现在Holroyd，Liggett和Romik关于Bootstrap渗流中有限大小尺度的研究中)，以证明Andrew所考虑的一族超几何Q-级数的尖端渐近展开。由于其范围，该研究程序具有广泛的应用潜力。一个潜在的主题是现代数论工具和技术的普遍性，其最著名的用途包括密码学和蜂窝通信。这项研究还将说明在高能物理(其中穿壁和黑洞是用模拟的theta函数来描述的)和生物‘增长’过程(其中细胞自动机的大规模行为由模形式的渐近展开来决定)中的应用。