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
|Effective start/end date||7/1/02 → 6/30/05|
- National Science Foundation: $138,906.00