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

数论和调和分析的领域是错综复杂地联系在一起的。 调和分析是数学的一个基本领域，应用于所有的STEM，包括数论。 数论研究整数中的模式，分析数论是指依赖于分析连续函数来理解离散对象的方法的集合。谐波分析提供了在离散和连续设置之间转换问题的桥梁。密西西比大学的数学系有一个活跃的解析数论小组，但在密西西比州没有调和分析师。通过这个奖学金，PI将获得谐波分析的研究专业知识，特别是当它适用于离散设置。PI将前往加州大学洛杉矶分校与陶教授合作并向他学习。陶有一个世界级的学生，博士后和游客的研究小组，与协作和合作学习的良好记录。PI将带一名博士生到洛杉矶，与UCLA分析组的学生学习和合作。该奖学金有可能对PI的职业生涯产生持久的影响，将加强密西西比大学的分析数论小组，并将加强研究生和本科生教育。这个研究基础设施改善轨道-4 EPSCoR研究员（RII轨道-4）项目将提供奖学金，以助理教授和培训研究生在密西西比大学（UM）。该奖学金将导致在解耦和限制理论的一般领域的两个或更多的合作研究项目。解析数论中的圆方法与调和分析中的解耦方法有着深刻的联系。在这个奖学金的第一年，PI将与陶和他的研究小组合作，通过研究某些缺乏膨胀-膨胀不变性的曲线的离散限制和解耦定理来探索这种联系。在第二年，在以前离散约束和解耦工作的基础上，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，并将与该国一些最强大的谐波分析师形成长期的合作伙伴关系。 PI将通过开发谐波分析的本科课程和研究生课程与密西西比大学社区分享这一新的专业知识。本科课程将是有用的数学专业计划进入工业，并将感兴趣的科学和工程专业的学生谁是学习信号处理和数据传输。该课程还将为PI和UM教师在其他STEM领域共同指导的跨学科论文项目创造机会。研究生课程将涵盖谐波分析的概念和技术，是相关的分析数论目前的研究。这两门课程都将为密西西比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。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.