Number Theory and Combinatorics

本提案中涉及的第一个主题是分区和概率。在这里，我们建议将Holroyd， Liggett和Romik在分区理论中开始的研究作为基础，以期获得(1)更好的渐近性，(2)更广泛的应用，以及(3)与Ramanujan的模拟θ函数的进一步联系。第二个主题，整个函数和罗杰斯-拉马努金恒等式，考察了埋藏在拉马努金遗失的笔记本中的两个真正惊人的公式所产生的进一步含义。第三个主题，恩格尔变换，建立在A.和J. Knopfmacher最初的一些惊人的展开定理的基础上。这个算法在q级数展开发挥重要作用的任何数学学科（包括统计力学）中都应该是有用的。第四个主题，冈田猜想，是一个已经开放了很多年的话题。最近在与Paule和Schneider的合作中，PI给出了这个猜想的q = 1情况的一个新的证明。这个新的证明清楚地表明，人们应该能够将其推广到一般的q。第五个主题，拉马努金和部分分式，是先前试图更好地理解拉马努金遗失笔记中一些更深奥的公式所提出的。在改进以前的工作方面取得了很大进展；希望这些方法可以进一步发展，以阐明模拟函数的一般理论。第六部分，也就是最后一部分，关注PI的研究与他改善教师教育的努力的交集。简而言之，很明显，组合理论（有序划分）可以在小学和中学教育中发挥更大的作用。本项目致力于研究分区和q级数中的问题，特别是那些同时(1)推进该数学分支的中心理论和(2)在其他数学和数学科学分支中具有巨大应用潜力的问题。关于分区和概率的部分尤其集中体现了这种哲学。这个主题已经应用于元胞自动机。关于恩格尔变换的工作研究了一种算法，其中有潜在的应用于统计物理问题。对整个函数、部分分式和冈田构造的研究应该开发出具有实际应用价值的方法，而不是当前的重点。第六部分，即最后一部分，关注研究与努力改善教师教育的交集。组合理论（有序分割）可以在中小学教育中发挥有益的辅助作用。

Project Summary for George Andrews The first topic covered in this propsoal is Partitions and Probability. Here it is proposed to ground the study begun by Holroyd, Liggett, and Romik in the theory of partitions with the hope of getting (1) better asymptotics, (2) broader applications, and (3) further connections with Ramanujan's mock theta functions. The second topic, Entire Functions and the Rogers-Ramanujan Identities, examines the further implications that arise from two truly amazing formulas that lay buried in Ramanujan's Lost Notebook. The third topic, Engel Transformations, builds upon some of the original amazing expansion theorems due to A. and J. Knopfmacher. This algorithm should be useful in any mathematical subject (including statistical mechanics) where q-series expansions play a substantial role. The fourth topic, The Okada Conjecture, is one that has been open for many years. Recently in joint work with Paule and Schneider, the PI gave a new proof of the q = 1 case of this conjecture. This new proof clearly suggests that one should be able to extend it to general q. The fifth topic, Ramanujan and Partial Fractions, was suggested by prior attempts to better understand some of the more recondite formulas in Ramanujan's Lost Notebook. Great progress has been made in improving previous work; it is hoped that these methods may be further developed to elucidate a general theory of mock theta functions. Section 6, the final section, concerns the intersection of the PI's research with his efforts to improve teacher education. Put succinctly, it is clear that the theory of compositions (ordered partitions) could play a much more substantial role in primary and secondary education. This project is devoted to problems in partitions and q-series, especially ones that simultaneously (1) advance the central theory of this branch of mathematics and (2) have substantial potential for application in other branches of mathematics and mathematical sciences. The section on partitions and probability especially epitomizes this philosophy. Applications of this topic have already been made to cellular automata. The work on Engel Transformations studies an algorithm wherein there are potential applications to problems in statistical physics. Work on entire functions, partial fractions and the Okada conejcture should develop methods with substantial applications beyond the current focus. Section 6, the final section, concerns the intersection of research with efforts to improve teacher education. The theory of compositions (ordered partitions) could well play a useful auxiliary role in primary and secondary education.