Number Theory and Allied Topics

9870060 Andrews该奖项支持Andrews教授和Brownawell各自项目的研究。安德鲁斯将专注于与分区理论和q级数相关的问题。最近，Andrews表明MacMahon的分割分析有很大的新发现潜力；这项工程将继续进行，并大大扩大。安德鲁斯与A. Berkovich合作，建立了q-三叉系数的贝利引理的类比。这里有几个很有希望的扩展，包括贝利引理和划分分析之间的联系。此外，Andrews最近发现了舒尔1926年划分定理的一个新方面，这应该允许更有效的证明和广泛推广他的定理（与Beesenrodt和Olsson联合）应用于模表示理论。布朗纳韦尔将重点讨论超越理论中的问题。他建议进一步研究德林菲尔德背景下的超越性质。在与L. Denis的联合工作中，他计划获得和推广关于可分离代数数的对数的线性无关性质的已知结果，涉及它的可分导数和准周期函数。他将研究Sinha在他的论文中如此引人注目地利用的情况，他还将研究将插值考虑应用于Carlitz指数函数的可能性。在其他项目中，布朗威尔将研究两个与微分方程和超越有关的问题。一个是西格尔提出的一个古老而被忽视的问题。另一个是涅斯捷连科最近研究中出现的一个新问题；一个肯定的答案将基于涅斯捷连科神奇定理的新证明，该证明基于协pi的代数独立性准则。Brownawell将研究Lojasiewicz不等式，该不等式将进一步改进Berenstein和Yger关于Nullstellensatz的令人印象深刻的工作。最后，他将开发一个多齐次的Nullstellens模型。本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，解决的问题是一个整数能被另一个整数整除的问题。它是数学中最古老的分支之一，人们为了纯粹的美学原因而追求了许多世纪。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各种应用领域不可或缺的工具。

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.