Applications of automorphic forms and hypergeometric q-series

本研究计划的主要目的是研究模和自同构形式以及超几何q级数的应用。这些应用包括数学和数学物理的不同领域的广泛选择，如整数分割理论，组合概率和马尔可夫过程，自举渗透模型，仿射李超代数，和（二次）Hurwitz类数。感兴趣的自同构对象包括模和Jacobi形式，以及模拟模和Jacobi形式，由于Borcherds、Bringmann、Bruinier、Funke、Ono、Zagier和Zwegers的工作，它们最近引起了很大的兴趣。模拟模形式特别有趣，因为它们与谐波质量形式的联系，以及它们作为神秘物体的著名历史，可以追溯到拉马努金和沃森。作为自同构形式与其他主题之间相互作用的一个例子，PI和Bringmann最近使用了避免间隙序列的组合概率界（首先出现在Holroyd， Liggett和Romik对自举渗透的有限尺寸缩放的研究中），以证明Andrews考虑的超几何q系列族的cusidal渐近展开式。由于其范围，该研究计划具有广泛应用的潜力。一个潜在的主题是现代数论的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（其中壁交叉和黑洞是由模拟θ函数描述的）和生物“生长”过程（其中元胞自动机的大规模行为是由模形式的渐近展开决定的）中的应用。

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).