Number Theory and Combinatorics

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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.