Automorphic Distributions, Summation Formulas, and Applications

米勒，斯蒂芬D.摘要标题：自守分布，求和公式和应用PI（米勒）的研究集中在自守形式的分析理论。 拟议的研究集中在自同构分布，一种技术，他一直在调查与威尔弗里德施密德（哈佛大学）。 他们希望利用这些分布来发展求和公式，类似于泊松和Voronoi的公式，从而可以使由模形式系数和加性扭曲加权的任意和二元论化。 PI计划研究分布和这些求和公式的各种应用。 许多人关注发展自守L-函数的新构造，特别是阿基米德和分歧计算可能的方法。 另一个是解析数论中的问题，如GL（n）上尖点形式的次凸性问题。 PI还计划改进一种通过颠倒求和公式来识别模形式的数值算法。自守形式的研究涉及现代数学研究的许多重要领域，包括数论、表示论、几何、分析和数学物理。Langlands通过L-函数揭示了自守形式之间的许多深刻而有趣的结构关系，这些关系在上述领域具有重要意义。作为一个例子，怀尔斯等人的工作证明了某些自守形式与求解正方形和立方体之间方程的古老问题之间的联系。本研究旨在从数学扩展微积分的分支分析和对称性的具体研究表示论出发，应用和发展自守形式和L-函数的新工具。目前自守形式和L-函数的应用表现在构造复杂的代码，使高速和安全的网上交易。

DMS-0301172Miller, Stephen D.AbstractTitle: Automorphic Distributions, Summation Formulas, and ApplicationsThe research of the PI (Miller) focuses on the analytic theory ofautomorphic forms. The proposed research concentrates on automorphicdistributions, a technique he has been investigating with Wilfried Schmid(Harvard University). They hope to use these distributions to developsummation formulas, similar to Poisson's and Voronoi's, whereby arbitrarysums weighted by modular form coefficients and additive twists can bedualized. The PI plans to investigate various applications of thedistributions and these summation formulas. Many concern developing newconstructions for automorphic L-functions, in particular approaches forwhich archimedean and ramified computations are possible. Another is toproblems in analytic number theory, such as the subconvexity problem forcusp forms on GL(n). The PI also plans to refine a numerical algorithmwhich identifies modular forms, by reversing summation formulas.The study of automorphic forms slices across many important areas ofmodern mathematical research, including number theory, representationtheory, geometry, analysis, and mathematical physics. Through L-functions,Langlands has conjectured many deep and interesting structuralrelationships between automorphic forms which have implications in theabove areas. As an example, the work of Wiles et al demonstrates the linkbetween certain automorphic forms and the ancient problem of solvingequations between squares and cubes. The proposed research aims to applyand develop new tools for automorphic forms and L-functions from analysis,which is the branch of mathematics expanding calculus, and representationtheory, the concrete study of symmetry. Current applications ofautomorphic forms and L-functions are manifest in constructing thesophisticated codes which enable high-speed and secure transactions overthe internet.