Some Problems in the Theory of Partitions and Q-series

在这个项目中，PI打算对分区理论和q级数中的一系列问题进行研究。在与Alexander Berkovich的合作中，PI首先提出了对Gollnitz的深度划分定理的四维扩展，并研究了其结果。第二个研究方向是在先前PI研究的基础上寻找新的加权划分恒等式。此外，PI打算对一个基本但未被利用的分区不变量进行系统研究，并获得各种重要q系列恒等式的多参数扩展。分割理论是用其他整数来表示整数，由于它与数学和相关学科的许多领域相互作用，是一个令人兴奋的研究领域。本课题是处理离散结构的数论和组合学的一部分。在20世纪80年代中期，在统计力学的研究中有了涉及分区的重大发现，这些发现具有重要的应用价值。最近，在物理共形场论的各种模型的研究中，分区已经被用于基本方法。近年来，随着强大的计算机代数软件包的出现，人们发现了许多新的组合恒等式。本提案的目标是发现新的和深入的配分恒等式，以及已知配分函数之间的新关系，这是由数学中不同分支和物理学中一些分支产生的问题所激发的。这些问题的进展将产生新的数学视角，并可能导致数学以外的重要应用。

In this project the PI intends to conduct research on a wide class of problems in the theory of partitions and q-series. In collaboration with Alexander Berkovich, the PI first proposes to obtain a four dimensional extension of a deep partition theorem of Gollnitz and study the consequences. A second direction of investigation is to search for new weighted partition identities based on earlier research of the PI. In addition, the PI intends to conduct a systematics study of a fundamental but unexploited partition invariant, and obtain multiparameter extensions of various important q-series identities.The theory of partitions which deals with the representations of whole numbers in terms of other whole numbers, is an exciting area of research owing to its interaction with many fields within mathematics and with allied disciplines. The subject is a part of the Number Theory and Combinatorics which deal with discrete structures. In the mid 1980s, major discoveries involving partitions were made in studies in statistical mechanics which had important applications. More recently, partitions have been used in fundamental ways in the study of various models in conformal field theory in physics. With the availability of powerful computer algebra packages in the last few years, many new combinatorial identities have been found. The goal of this proposal is to discover new and deep partition identities, as well as new relationships among well known partition functions, motivated by questions arising from different branches within mathematics, and some from physics. Progress on these problems would yield new mathematical perspectives and might lead to significant applications outside of mathematics.