Number Theory and Combinatorics

本文主要研究q系列和分区中的问题。这项工作有五个独立的部分。第一部分考虑与李代数表示的构造应用相关的研究。接下来，研究者着眼于与数论中的特殊问题有关的新的q系列方法。第三部分讨论了Omega软件包（http://www.uni-linz.ac.at/research/combinat/risc/software/Omega/）的应用程序，该软件包正在由研究人员与林茨的同事合作开发。后一节的重点是多维分区。第四部分致力于对贝利链的研究，并考虑了研究者的最新发现如何导致这一概念的新应用。本文最后考虑了分区理论中三个尚未解决的主要问题：(1)Friedman-Joichi-Stanton猜想，(2)Borwein猜想和(3)Okada猜想。这三个猜想中的每一个都存在了一段时间。这个提议的主题可以简洁地说：建立从分区和q系列（两个本质上深刻而迷人，但有时相当内向的话题）到数学和科学的几个分支的桥梁。前两部分致力于将这项工作与数学的两个分支表示论和数论联系起来；在每个实例中，很明显，这种交互不仅会丰富对象字段，而且还会为分区和q系列提供新的见解。欧米茄包的工作有很大的潜力。在这里，研究者和他的合作者发现了许多研究发现从不可想象到容易实现的例子。对多维分区的可能应用应该会导致对组合学的深入了解，并有可能导致对物理学组合方面的深入了解。过去关于贝利链的工作对物理学中的统计力学产生了深远的影响。这种方法越先进，我们就越有可能期待这些互惠互利的应用继续下去。关于三个尚未解决的问题的最后一节，起初似乎是一个纯粹的内部研究。然而，正如过去经常发生的那样，每当发现解决真正困难问题的新方法时，几乎总是会溢出到重要的应用程序中。

This proposal focuses on problems in q-series and partitions. There are five separate parts of this work. The first part considers research tied to applications of the construction of representations of Lie algebras. Next the investigator looks at new q-series methods related to special problems in number theory. The third part discusses applications of the Omega software package (http://www.uni-linz.ac.at/research/combinat/risc/software/Omega/) which is being developed by the investigator in collaboration with colleagues at Linz. The focus in this latter section is on mutli-dimensional partitions. The fourth section is devoted to the study of Bailey chains and a consideration ofhow recent discoveries of the investigator may lead to new applications of this concept. The proposal concludes with consideration of three major unsolvedproblems in the theory of partitions: (1) the Friedman-Joichi-Stanton conjecture, (2) the Borwein conjecture and (3) the Okada conjecture. Each of these three conjectures has been around for some time.The theme of this proposal put succinctly might be: Building bridges from partitions and q-series (two intrinsically deep and charming but sometimes rather introverted topics) to several branches of mathematics and science. The first two sections are devoted to relating this work to representation theory and number theory, two branches of mathematics; in each instance, it is clear that this interaction will not only enrich the object fields, but also will provide new insights for partitions and q-series. The work on the Omega package has great potential. Here the investigator and his collaborators have found numerous instances where research discoveries have gone from being unthinkable to easily reached. The possible applications to multi-dimensional partitions should lead to insights in combinatorics and, hopefully, the combinatorial aspects of physics. The work on Bailey chains in the past has had profound impact on statistical mechanics in physics. The more this method is advanced, the more we may expect these mutually beneficial applications to continue. The final section on three unsolved problems appears, at first, to be a purely internal study. However, as has often happend in the past, whenever new methods are discovered to solve really hard problems, there is almost always a spillover into vital applications.