Some Problems in the Theory of Partitions and Q-series

在这个项目中，PI打算对分拆理论和Q-级数中的一大类问题进行研究。与Alexander Berkovich合作，PI首次提出将Gollnitz的一个深分拆定理推广到四维，并研究其结果。第二个研究方向是在PI的早期研究的基础上寻找新的加权划分恒等式。此外，PI还打算对一个基本的但尚未开发的分拆不变量进行系统研究，并获得各种重要的Q-级数恒等式的多参数扩展。分拆理论处理整数用其他整数表示的问题，由于它与数学中的许多领域和相关学科的相互作用，是一个令人兴奋的研究领域。这门学科是数论和组合学的一部分，后者涉及离散结构。在20世纪80年代中期，在统计力学的研究中取得了涉及划分的重大发现，这具有重要的应用。最近，在物理学中共形场理论中的各种模型的研究中，已经基本地使用了分割。随着最近几年功能强大的计算机代数软件包的出现，许多新的组合恒等式被发现。这项提议的目标是发现新的和深层次的划分身份，以及众所周知的划分函数之间的新关系，这些问题来自数学中不同分支的问题，以及一些来自物理的问题。在这些问题上的进展将产生新的数学观点，并可能导致在数学之外的重大应用。

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.