Summation Formulae and Triple Product L-functions in Higher Rank

该奖项支持的研究旨在从根本上推进我们对朗兰兹功能性猜想的理解。 朗兰兹函数性猜想是现代数学中一个深刻的统一力量。 虽然在很大程度上是开放的，但即使是目前已知的结果也会在数论、表示论、代数几何甚至数学物理中产生涟漪。研究的基本对象是自守表示。 自守表示是使用局部对称空间定义的，局部对称空间是高度对称的空间，可以被认为是任意维的鼓。 鼓上的每一个声音都是纯音的叠加，自守表示类似于这些纯音。 这些都是非常复杂的物体。每个局部对称空间都有一个相对简单的不变量，称为L-群，而函子性猜想预言，当一个人可以联系L-群时，他可以联系（复杂得多的）自守表示。PI还将培养这一领域的研究生，并参与从高中到博士后的各个层次的指导活动。使用L-群之间的关系定义称为L-函数的函数，证明L-函数是“好”的就足以证明朗兰兹函子性。 这个项目将开发新的技术来证明L-函数是好的。 最古老的工具，可用于证明L-函数是好的是泊松求和公式熟悉傅立叶分析，信号处理等PI和B。刘最近发现的推广的泊松求和公式是第一次。更具体的研究目标是推广和利用这些推广，目的是研究更高秩的三重积L-函数。 使用Cogdell和Piatetski-Shapiro开发的匡威理论，朗兰兹函数性的许多案例可以简化为三重乘积L函数的研究，因此这具有根本的重要性。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

The research supported under this award is aimed at fundamentally advancing our understanding of the Langlands functoriality conjecture. The Langlands functoriality conjecture is a profound unifying force in modern mathematics. Though largely open, even what is currently known has consequences that ripple throughout number theory, representation theory, algebraic geometry, and even mathematical physics. The basic objects of study are automorphic representations. Automorphic representations are defined using locally symmetric spaces, which are highly symmetric spaces that can be thought of as arbitrary dimensional drums. Every sound on a drum is a superposition of pure tones and automorphic representations are analogous to these pure tones. These are very complicated objects. Every locally symmetric space has a comparatively simple invariant called an L-group, and the functoriality conjecture predicts that when one can relate L-groups one can relate (the far more complicated) automorphic representations. The PI will also train graduate students in this area of research and is involved in mentoring activities at all levels from high school through to post-doctoral.Using relations between L-groups one defines functions called L-functions, and proving L-functions are "nice" is enough to prove Langlands functoriality. This proejct will develop new techniques to prove that L-functions are nice. The oldest tool available for proving L-functions are nice is the Poisson summation formula familiar from Fourier analysis, signal processing, etc. The PI and B. Liu have recently discovered generalizations of the Poisson summation formula that are the first of their kind. The more particular goal of the research is to extend and exploit these generalizations with the aim of studying triple product L-functions in higher rank. Using converse theory as developed by Cogdell and Piatetski-Shapiro many cases of Langlands functoriality can be reduced to the study of triple product L-functions so this is of fundamental importance.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.