Number Theory and Combinatorics

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.