Number Theory and Combinatorics

Project: Research project

Project Details

Description

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 unsolved

problems 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.

StatusFinished
Effective start/end date7/1/026/30/05

Funding

  • National Science Foundation: $138,906.00

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