RII Track-4:NSF: From Analytic Number Theory to Harmonic Analysis

数论和调和分析这两个领域有着错综复杂的联系。调和分析是数学的一个基本领域，在STEM的所有领域都有应用，包括数论。数论研究整数中的模式，而解析数论指的是一组依赖于分析连续函数来理解离散对象的方法。调和分析提供了在离散和连续设置之间转换问题的桥梁。密西西比大学数学系有一个活跃的解析数论小组，但密西西比州没有调和分析员。通过这一奖学金，PI将获得谐波分析方面的研究专业知识，特别是当它应用于离散设置时。国际和平协会将前往加州大学洛杉矶分校，与陶渊明教授合作并向其学习。陶拥有一支由学生、博士后和来访者组成的世界级研究团队，在合作和合作学习方面有着良好的记录。PI将把一名博士生带到洛杉矶，向加州大学洛杉矶分校分析小组的学生学习并与他们合作。这一奖学金有可能对PI的职业生涯产生持久的影响，将加强密西西比大学的解析数论小组，并将加强研究生和本科教育。研究基础设施改进Track-4 EPSCoR研究人员(RII Track-4)项目将为密西西比大学(UM)的一名助理教授提供奖学金，并为一名研究生提供培训。这一奖学金将导致两个或更多在脱钩和限制理论的一般领域的合作研究项目。解析数论中的圆法与调和分析中的解耦法有着深刻的联系。在这个奖学金的第一年，PI将与陶和他的研究小组合作，通过研究某些曲线的离散限制和解耦定理来探索这种联系，这些曲线缺乏平移-膨胀不变性。在第二年，在之前关于离散约束和解耦的工作的基础上，合作将转向连续约束理论领域。通过这一系列的合作，PI将获得一个新的研究领域，并将与国内一些最强大的和声分析师形成长期的合作伙伴关系。PI将通过开发一门本科生课程和一门谐波分析研究生课程，与密西西比大学社区分享这一新的专业知识。这门本科课程将对计划进入工业的数学专业的学生有用，对学习信号处理和数据传输的理工科学生也会有兴趣。这门课程还将为PI和UM教员在其他STEM领域共同建议的跨学科论文项目创造机会。研究生课程将涵盖与当前解析数论研究相关的调和分析概念和技术。这两门课程都将为密西西比州的学生提供宝贵的工具，使他们成为更有效的数学家、科学家和工程师。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The fields of number theory and harmonic analysis are intricately connected. Harmonic analysis is a fundamental area of mathematics with applications across all of STEM, including number theory. Number theory examines patterns in the integers, and analytic number theory refers to a collection of methods that rely on analyzing continuous functions to understand discrete objects. Harmonic analysis provides the bridge that converts problems between the discrete and continuous settings. The Department of Mathematics at the University of Mississippi has an active analytic number theory group, but there are no harmonic analysts in the state of Mississippi. Through this fellowship, the PI will gain research expertise in harmonic analysis, especially as it applies to discrete settings. The PI will travel to the University of California, Los Angeles to collaborate with and learn from Professor Terence Tao. Tao has a world-class research group of students, postdocs, and visitors, with a strong record of collaboration and cooperative learning. The PI will bring one PhD student to Los Angeles to learn from and collaborate with the students in the UCLA analysis group. This fellowship has potential to have a lasting impact on the PI’s career, will strengthen the analytic number theory group at the University of Mississippi, and will enhance graduate and undergraduate education. This Research Infrastructure Improvement Track-4 EPSCoR Research Fellows (RII Track-4) project would provide a fellowship to an Assistant Professor and training for a graduate student at the University of Mississippi (UM). This fellowship will result in two or more collaborative research projects in the general area of decoupling and restriction theory. There is a deep connection between the circle method in analytic number theory and the method of decoupling in harmonic analysis. During the first year of this fellowship, the PI will collaborate with Tao and his research group to explore this connection by investigating discrete restriction and decoupling theorems for certain curves which lack translation-dilation invariance. In the second year, building on the previous work with discrete restriction and decoupling, the collaboration will pivot into the area of continuous restriction theory. Through this sequence of collaborations, the PI will gain a new research area and will form long-lasting collaborative partnerships with some of the strongest harmonic analysts in the country. The PI will share this new expertise with the University of Mississippi community by developing an undergraduate course and a graduate course in harmonic analysis. The undergraduate course will be useful to mathematics majors planning to go into industry, and will be of interest to science and engineering students who are studying signal processing and data transfers. The course will also create opportunities for interdisciplinary thesis projects co-advised by the PI and UM faculty in other STEM fields. The graduate course will cover harmonic analysis concepts and techniques that are relevant to current research in analytic number theory. Both courses will equip Mississippi students with valuable tools to become more effective mathematicians, scientists, and engineers.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.