Class Groups, Character Sums, and Oscillatory Integrals

这个项目将研究数论和调和分析数学领域的中心问题和猜想。这项研究将阐明基本数学对象的性质，并将开发未来可供其他数学家应用的新技术。此外，首席研究员莉莲·皮尔斯博士将加强科学知识的传播，扩大对STEM研究和学习的参与。皮尔斯博士将在进行原创研究和获得数学博士学位的过程中资助和指导受训人员。该奖项将支持两个会议，这两个会议将帮助女性和非二进制学生全面参与美国的数学博士项目，一个调和分析暑期班，以及创新的新期刊《基本数论》。皮尔斯博士的研究将集中在解析数论中的公开问题，以及受算术观点启发的分析项目。这些问题围绕三个主题展开。第一个项目将使用特征标和、筛子和L函数的方法来研究关于数域的类群的一个著名的猜想。第二个项目将研究短混合特征和，并将其应用于理解L函数和改进计算丢番图方程整数解的筛法。第三个项目将研究平方函数、偏微分方程的收敛问题和最大调制振荡积分算子。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。

This project will investigate central questions and conjectures in the mathematical fields of number theory and harmonic analysis. The research will elucidate properties of fundamental mathematical objects, and will develop new techniques that can be applied in the future by other mathematicians. In addition, the principal investigator, Dr. Lillian Pierce, will enhance the transmission of scientific knowledge and broaden participation in STEM research and learning. Dr. Pierce will fund and mentor trainees in the process of performing original research and obtaining PhD’s in mathematics. This award will support two conferences that aid the full participation of women and non-binary students in US-based math PhD programs, a summer school in harmonic analysis, and the innovative new journal “Essential Number Theory.”Dr. Pierce’s investigations will focus on open questions in analytic number theory, as well as projects in analysis that are motivated by an arithmetic viewpoint. These questions center on three themes. The first project will investigate a well-known conjecture on class groups of number fields, using methods involving character sums, sieves, and L-functions. The second project will investigate short mixed character sums, with applications to understanding L-functions and improving sieve methods for counting integral solutions to Diophantine equations. The third project will investigate square functions, convergence questions in partial differential equations, and maximally modulated oscillatory integral operators.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.