Number Theory and Combinatorics

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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.