Topics in Analytic Number Theory and Additive Combinatorics
解析数论和加法组合学主题
基本信息
- 批准号:2200565
- 负责人:
- 金额:$ 16.85万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2022
- 资助国家:美国
- 起止时间:2022-07-01 至 2025-06-30
- 项目状态:未结题
- 来源:
- 关键词:
项目摘要
This is an interdisciplinary project connecting number theory and combinatorics. One central theme in analytic number theory is to understand the distribution of primes numbers, a topic that has important applications in cryptography, coding theory, and financial security. It is a field that is currently experiencing intensive progress, in sometimes unexpected directions. In recent years, many important classical questions have seen spectacular advances based on new techniques; conversely, methods developed in analytic number theory have led to the solution of striking problems in other fields. The main goal of this research project is to make progress on the distributional properties of primes and other related arithmetic objects, by developing further the classical analytic tools and the more modern additive combinatorial tools, and by understanding how to effectively combine these two different tools together. It is hoped that the need for stronger results to cater for applications in analytic number theory will drive the development of additive combinatorics, and vice versa. The grant will also be used to help train graduate students in the area.More specifically, the PI will continue his work surrounding additive problems with primes. These works will involve developing both analytic tools and combinatorial tools. On the analytic side, the PI will continue to investigate the theory of Gowers norms for multiplicative functions and for the von-Mangoldt function, which are higher-order extensions to classical exponential sum estimates involving these functions. On the combinatorial side, the PI expects to develop tools from additive combinatorics for locating arithmetic structures within dense sets of integers and discover new mechanisms for using them to exhibit such structures in the primes. The additive combinatorial ideas behind these tools, such as pseudorandomness and Gowers uniformity norms, have a strong link with topics in theoretical computer science such as extractors and expanders, property testing, and error-correcting codes. The PI plans to explore these links in this project. This project is jointly funded by the Algebra and Number Theory program, the Established Program to Stimulate Competitive Research (EPSCoR), and the Combinatorics program.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
这是一个连接数论和组合学的跨学科项目。解析数论的一个中心主题是理解素数的分布,这个主题在密码学、编码理论和金融安全中有着重要的应用。这是一个目前正在经历密集进展的领域,有时会出现意想不到的方向。近年来,许多重要的经典问题在新技术的基础上取得了惊人的进展;相反,在解析数论中发展起来的方法也导致了其他领域中引人注目的问题的解决。本研究项目的主要目标是通过进一步发展经典分析工具和更现代的加法组合工具,并了解如何有效地将这两种不同的工具结合在一起,在素数和其他相关算术对象的分布性质方面取得进展。人们希望,为了满足分析数论应用的需要,需要更强的结果,这将推动加性组合学的发展,反之亦然。这笔拨款还将用于帮助培训该地区的研究生。更具体地说,PI将继续围绕素数的加性问题进行研究。这些工作将包括开发分析工具和组合工具。在分析方面,PI将继续研究乘法函数和von-Mangoldt函数的Gowers范数理论,这是对涉及这些函数的经典指数和估计的高阶扩展。在组合方面,PI期望从加法组合学中开发工具,用于在密集整数集中定位算术结构,并发现使用它们在素数中展示这种结构的新机制。这些工具背后的附加组合思想,如伪随机性和高尔斯均匀性规范,与理论计算机科学中的主题有很强的联系,如提取器和扩展器、性能测试和纠错码。PI计划在本项目中探索这些联系。该项目由代数和数论项目、刺激竞争性研究的既定项目(EPSCoR)和组合学项目共同资助。该奖项反映了美国国家科学基金会的法定使命,并通过使用基金会的知识价值和更广泛的影响审查标准进行评估,被认为值得支持。
项目成果
期刊论文数量(0)
专著数量(0)
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会议论文数量(0)
专利数量(0)
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Xuancheng Shao其他文献
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{{ truncateString('Xuancheng Shao', 18)}}的其他基金
Topics in Analytic Number Theory and Additive Combinatorics
解析数论和加法组合学主题
- 批准号:
1802224 - 财政年份:2018
- 资助金额:
$ 16.85万 - 项目类别:
Standard Grant
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