Mathematical Sciences: Number Theory and Allied Topics

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9501101: Andrews Abstract. This award supports the research of Professors George Andrews and W. Dale Brownawell in combinatorics, number theory and commutative algebra. One main theme of this research is to develop new methods in q-series and hypergeometric series to further discoveries in the theory of partitions, number theory, representation theory and the applications of random graphs in computer science. A second direction is to study an algorithmic effective Nullstellensatz, isogeny of Drinfeld modules, independence criteria in diophantine rings, and zero estimates for solutions of differential equations. This research is in the general areas of combinatorics and number theory, and also involves the interaction of these areas with commutative algebra and algebraic geometry. Combinatorics attempts to find efficient methods to study how discrete collections of objets can be arranged. The behavior of discrete systems is extremely important to modern communications. For example, the design of large networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research. 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.