Number Theory and Allied Topics

9870060 Andrews This award supports research for separate projects by Professor Andrews and Brownawell. Andrews will focus on problems related to the theory of partitions and q-series. Recently, Andrews showed that MacMahon's Partition Analysis has great potential for new discoveries; this project will be continued and greatly expanded. In collaboration with A. Berkovich, Andrews established an analog of Bailey's Lemma for q-trinomial coefficients. There are several promising possible extensions here that will be explored including ties between Bailey's Lemma and Partition Analysis. In addition, Andrews has recently found a new aspect of Schur's 1926 partition theorem which should allow a more efficient proof and vast generalization of his theorems (joint with Beesenrodt and Olsson) for application to modular representation theory. Brownawell will focus on problems in transcendence theory. He proposes to further investigate transcendence properties in the Drinfeld setting. In joint work with L. Denis, he plans to obtain and extension of known results on the linear independence properties of the logarithm of a separably algebraic number involving the divided derivatives of it and of quasi-periodic functions. He will look at the situation that Sinha exploited so strikingly in his thesis, and he will also look into the possibility of applying interpolation considerations for the Carlitz exponential functions. In other projects, Brownawell will look at two problems relating differential equations and transcendence. One is an old and neglected question of Siegel's. The other is a new question that arose in recent work of Nesterenko; a positive answer here would base a new proof of Nesterenko's marvelous theorems on a criterion of algebraic independence due to the co-PI. Brownawell will investigate a Lojasiewicz inequality that will improve even further the current impressive work of Berenstein and Yger on the Nullstellensatz. Finally, he will develop a multi-homogeneous Nullstellens atz. This research falls into the general mathematical field of number theory. 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.