REU Site: Computational Number Theory

REU 网站：计算数论

• 批准号: 2349174
• 负责人: Hui Xue
• 金额: $ 28.84万
• 依托单位: Clemson University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-11-01 至 2027-10-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2349174&HistoricalAwards=false
• 关键词: REU; Site; Computational; Number; Theory

Eight students across the country will participate in an eight-week research experience in computational number theory at Clemson University each year of this project. The goal of this program is to help students attain a higher level of independence in mathematical research by having them take part in significant and interesting research projects. Participants will be introduced to various tools, techniques, and problems from number theory and will work on important and often difficult problems that are suitable for undergraduate work. This program will not only provide the participants with the opportunities to broaden their knowledge in their research area but will also give participants the opportunity to become better expositors of their research. This will be accomplished through student lectures during and at the end of the program that will include two presentations in different formats. The PIs will organize an annual REU conference that will enable the students to interact with and learn from students in other REU programs in the region. The conference will provide the students with a valuable opportunity to deliver their first professional talk in a friendly atmosphere and will also help to disseminate the results obtained by the REU students. This project is jointly funded by the Mathematical Sciences Research Experiences for Undergraduates Sites program and the Established Program to Stimulate Competitive Research program.The theory of modular forms plays an important role in modern number theory, such as Andrew Wiles' proof of Fermat's Last Theorem. In this program, various problems in modular forms will be introduced. These problems will offer a blend of computational investigation with the theoretical pursuit of fundamental problems in modular forms or, more generally, in number theory. The problems are specifically chosen so that the participants will be able to begin investigations almost immediately on computational aspects of the projects, giving them an opportunity to spend the entire time at Clemson working on meaningful research. Potential research projects include but are not limited to studying the distribution of zeros of certain modular forms or period polynomials, investigating properties of higher coefficients of Hecke polynomials, and analyzing traces of Hecke operators on certain types of modular forms. Many of these problems are natural extensions or continuations of the results obtained in the previous REUs. More information can be found at the REU website: https://huixue.people.clemson.edu/REU.html.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.

该项目每年都会有八名全国各地的学生在克莱姆森大学参加为期八周的计算数论研究。该计划的目标是通过让学生参与重要且有趣的研究项目来帮助他们在数学研究中获得更高水平的独立性。参与者将了解数论中的各种工具、技术和问题，并将解决适合本科工作的重要且通常困难的问题。该计划不仅将为参与者提供扩大其研究领域知识的机会，而且还将为参与者提供成为更好的研究阐释者的机会。这将通过课程期间和课程结束时的学生讲座来完成，其中包括两次不同格式的演讲。 PI 将组织年度 REU 会议，使学生能够与该地区其他 REU 项目的学生互动并向他们学习。此次会议将为学生们提供一个在友好的氛围中进行首次专业演讲的宝贵机会，也将有助于传播 REU 学生所取得的成果。该项目由本科生数学科学研究经验项目和刺激竞争性研究既定项目共同资助。模形式理论在现代数论中发挥着重要作用，例如Andrew Wiles对费马大定理的证明。在本程序中，将以模块化的形式介绍各种问题。这些问题将把计算研究与模形式或更普遍的数论中的基本问题的理论追求结合起来。这些问题是经过专门选择的，以便参与者几乎能够立即开始对项目的计算方面进行调查，使他们有机会在克莱姆森大学进行有意义的研究。潜在的研究项目包括但不限于研究某些模形式或周期多项式的零点分布、研究赫克多项式的较高系数的性质以及分析赫克算子在某些类型的模形式上的踪迹。其中许多问题是先前 REU 中获得的结果的自然延伸或延续。更多信息可在 REU 网站上找到：https://huixue.people.clemson.edu/REU.html。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查进行评估，被认为值得支持标准。