CAREER: Local-global phenomena and sieves in thin orbits

这一提议主要涉及希尔伯特第11问题的新变体，即在非常规环境中建立局部到全局的原则。具体来说，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圈。PI建议将现有的工具扩展到许多其他情况，包括阿波罗3-垫圈和Soddy球填料，长期目标是更好地理解应用于这种薄轨道的多线性形式的圆方法和指数和界可以成功的情况。在甚至“几乎”局部-全局陈述似乎超出当前技术范围的情况下，PI将开发新的指数和界，以提高可以从仿射筛产生的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对应于真实谱。与拟议的研究项目相结合的是许多教育和推广组件。PI将继续通过研究生为高中提供建议，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公众授课，特别是针对科学之路项目中的初中和高中学生。该提案还将支持纽黑文地区贫困中学生参加MathCounts竞赛。通过这些途径，PI将能够使传统上代表性不足的群体接触到前沿研究。该奖项由代数和数论以及分析项目共同资助。

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.