Number Theory and Allied Topics

9870060安德鲁斯该奖项支持安德鲁斯教授和布朗威尔教授为不同项目所做的研究。安德鲁斯将专注于与分拆理论和Q-级数相关的问题。最近，安德鲁斯表明，麦克马洪的分区分析具有巨大的新发现潜力；该项目将继续并大大扩展。在与A.Berkovich的合作中，Andrews建立了类似于Bailey关于q-三项式系数的引理。这里有几个可能的扩展，包括Bailey引理和划分分析之间的联系。此外，Andrews最近在Schur的1926年的划分定理中发现了一个新的方面，它应该允许更有效的证明和他的定理的广泛推广(与Beesenrodt和Olsson一起)，以应用于模表示理论。布朗纳威尔将专注于超越论中的问题。他建议在德恩菲尔德的背景下进一步研究超越性属性。在与L.Denis的合作中，他计划得到并推广关于涉及可分代数的导数的可分代数数和拟周期函数的对数的线性无关性的已知结果。他将研究辛哈在他的论文中如此引人注目地利用的情况，他还将研究将内插考虑应用于Carlitz指数函数的可能性。在其他项目中，布朗纳韦尔将研究两个与微分方程和超越性相关的问题。一个是Siegel的一个古老而被忽视的问题，另一个是Nester enko最近工作中出现的一个新问题；在这里，一个肯定的答案将基于由co-PI产生的代数无关性准则来证明Nester enko的奇妙定理。Brownawell将研究一个Lojasiewicz不等式，它将进一步改进Berenstein和Yger目前在Nullstellensatz上令人印象深刻的工作。最后，他将开发一个多重齐次Nullstellens atz。这项研究属于数论的一般数学领域。数论的历史根源在于对整数的研究，它解决了一些问题，比如一个整数被另一个整数整除的问题。它是数学中最古老的分支之一，出于纯粹的美学原因，人们追寻了许多个世纪。然而，在过去的半个世纪里，它已经成为数据传输和处理以及通信系统等领域的各种应用中不可或缺的工具。

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.