Number Theory and Allied Topics

小行星9870060 该奖项支持安德鲁斯教授和布朗纳韦尔单独项目的研究。安德鲁斯将专注于与分区和q系列理论相关的问题。最近，安德鲁斯表明，麦克马洪的分区分析有很大的潜力，新的发现，这个项目将继续下去，大大扩大。与A合作。Berkovich，Andrews建立了q-三项式系数的Bailey引理的类似物。有几个有前途的可能的扩展，将在这里探讨，包括贝利引理和分区分析之间的联系。此外，安德鲁斯最近发现了一个新的方面舒尔的1926年分割定理，应允许更有效的证明和广泛推广他的定理（联合Beesenrodt和奥尔森）的应用模块化表示理论。布朗纳威尔将集中在超越理论的问题。他建议进一步研究超越性质的德林费尔德设置。在与L.丹尼斯，他计划获得和扩展的已知结果的线性独立性质的对数的一个可分代数数涉及除衍生物，它和准周期函数。他将着眼于这种情况下，辛哈利用如此惊人的在他的论文，他还将探讨的可能性，适用插值考虑的Carlitz指数函数。在其他项目中，Brownawell将着眼于两个有关微分方程和超越的问题。一个是西格尔提出的一个古老而被忽视的问题。另一个是一个新的问题，出现在最近的工作涅斯捷连科;一个积极的答案在这里将根据一个新的证明涅斯捷连科的了不起的定理的标准代数独立由于共同PI。Brownawell将研究Lojasiewicz不等式，该不等式将进一步改进Berenstein和Yger关于Nullstelleneux的当前令人印象深刻的工作。最后，他将开发一个多齐次Nullstellens atz。 本研究属于一般数学领域的数论研究福尔斯。数论有其历史根源，在研究整个数字，解决这样的问题，如那些处理整除一个整数由另一个。它是数学中最古老的分支之一，几个世纪以来，人们纯粹出于美学的原因而追求它。然而，在过去的半个世纪，它已成为一个不可或缺的工具，在不同的应用领域，如数据传输和处理，通信系统。

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.