CAREER: Local-global phenomena and sieves in thin orbits

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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.