Combinatorial, Differential and Modular Partition Problems

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9870052GarvanThis award supports the research of Frank Garvan in number theory and combinatorics. In particular three partition problems will be studied. At the most basic level partition theory deals with the problem of counting the number of ways a number can be written as a sum of smaller numbers. The theory of partition has had important connections with other branches of number theory such as the theory of modular functions. It has also had important applications to physics particularly statistical mechanics. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems. Modular functions are functions with highly symmetric properties. One main goal of this work is to apply the classical theory of modular functions in a non-standard way to some problems in partition theory.