Some Problems in the Theory of Partitions and Q-series

University Of Florida, Gainesville FL

Investigators

Abstract

In this project the PI intends to conduct research on a wide class of problems in the theory of partitions and q-series. In collaboration with Alexander Berkovich, the PI first proposes to obtain a four dimensional extension of a deep partition theorem of Gollnitz and study the consequences. A second direction of investigation is to search for new weighted partition identities based on earlier research of the PI. In addition, the PI intends to conduct a systematics study of a fundamental but unexploited partition invariant, and obtain multiparameter extensions of various important q-series identities. The theory of partitions which deals with the representations of whole numbers in terms of other whole numbers, is an exciting area of research owing to its interaction with many fields within mathematics and with allied disciplines. The subject is a part of the Number Theory and Combinatorics which deal with discrete structures. In the mid 1980s, major discoveries involving partitions were made in studies in statistical mechanics which had important applications. More recently, partitions have been used in fundamental ways in the study of various models in conformal field theory in physics. With the availability of powerful computer algebra packages in the last few years, many new combinatorial identities have been found. The goal of this proposal is to discover new and deep partition identities, as well as new relationships among well known partition functions, motivated by questions arising from different branches within mathematics, and some from physics. Progress on these problems would yield new mathematical perspectives and might lead to significant applications outside of mathematics.