CAREER: Local-global phenomena and sieves in thin orbits

这一建议主要涉及希尔伯特第11个问题的新变种，即在非传统环境中建立局部到全球的原则。具体地说，PI将研究由薄轨道表示的整数，一个典型的例子是积分阿波罗垫片。国际和平研究所建议将现有工具扩展到许多其他情况，包括阿波罗3-型垫圈和索迪球填料，其长期目标是更好地了解在什么情况下适用于如此薄的轨道的圆法和多线性形式的指数和界可以成功。在即使是“几乎”的局部-全局陈述似乎都不能满足当前技术的情况下，PI将开发新的指数和界来改进可以从仿射筛子产生的R-几乎素数的数量。PI还将开发数值算法来探索无限体积双曲流形的拉普拉斯谱，并努力严格证明数值观察到的特征值实际上与真实谱相对应。与拟议的研究项目整合在一起的是许多教育和推广部分。PI将继续为高中至研究生提供建议，并组织研讨会、会议和会议。耶鲁大学将为非数学专业的学生开发一门新课程，目标是向更广泛的受众普及数学，包括讨论最近的研究活动。PI将通过纽黑文和国家事务办公室以及耶鲁大学的科学星期六向普通公众讲课，专门面向科学之路项目的初中生和高中生。这项提案还将支持纽黑文地区贫困中学生参加MathCounts比赛。通过这样的途径，PI将能够让传统上代表不足的群体接触到尖端研究。该奖项由代数和数论以及分析程序共同资助。

This proposal is concerned largely with novel variants of Hilbert's 11th Problem, namely establishing local-to-global principles in unconventional settings. Specifically, the PI will study integers represented by thin orbits, a quintessential example being integral Apollonian gaskets. The PI proposes to extend existing tools to many other situations, including Apollonian 3-gaskets and Soddy sphere packings, with the long-term goal of better understanding circumstances under which the circle method and exponential sum bounds for multi-linear forms, applied to such thin orbits, can succeed. In situations where even an "almost" local-global statement seems out of reach of current technology, the PI will develop new exponential sum bounds to improve the number of R-almost primes which can be produced from an Affine Sieve. The PI will also develop numerical algorithms to explore the Laplace spectrum of infinite volume hyperbolic manifolds, and strive to rigorously certify that numerically observed eigenvalues actually correspond to true spectra.Integrated with the proposed research projects are numerous educational and outreach components. The PI will continue to advise high school through graduate students, and organize seminars, meetings, and conferences. A new course at Yale will be developed for non-math majors, with the goal of popularizing mathematics to broader audiences, including discussions of recent research activity. The PI will give lectures through the Office of New Haven and State Affairs, and Science Saturdays at Yale, to the general public, specifically geared for middle and high school students in the Pathways to Science program. This proposal will also support the attendance of underprivileged New Haven area middle school students to the MathCounts competition. Through such avenues, the PI will be in a position to bring traditionally under-represented groups into contact with cutting-edge research.This award is cofunded by the Algebra and Number Theory and the Analysis programs.