REU Site: Computational Number Theory

全国各地的八名学生每年将在克莱姆森大学参加为期八周的计算数论研究经验。该计划的目标是通过让学生参与重要而有趣的研究项目，帮助他们在数学研究中获得更高水平的独立性。参与者将被介绍到数论的各种工具、技术和问题，并将研究适合本科工作的重要且通常困难的问题。该计划不仅为参与者提供了在研究领域扩大知识的机会，而且还为参与者提供了成为更好的研究说明者的机会。这将通过学生在课程期间和课程结束时的讲座来完成，其中包括两种不同格式的演讲。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。会议将为学生提供一个宝贵的机会，在友好的气氛中发表他们的第一次专业演讲，并有助于传播REU学生取得的成果。本项目由“本科生数学科学研究经验”项目和“促进竞争性研究的既定项目”共同资助。模形式理论在现代数论中占有重要地位，如安德鲁·怀尔斯对费马大定理的证明。在这个节目中，将介绍模块化形式的各种问题。这些问题将提供一个混合的计算调查与理论追求的基本问题，在模的形式，或更一般地说，在数论。这些问题是经过特别挑选的，这样参与者就可以几乎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的整个时间里从事有意义的研究。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分布，研究Hecke多项式的高系数性质，分析Hecke算子在某些模形式上的迹线。这些问题中的许多都是前面reu中得到的结果的自然扩展或延续。更多信息可以在REU网站上找到：https://huixue.people.clemson.edu/REU.html.This该奖项反映了NSF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。

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.