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泛函猜想的理解。朗兰兹泛函猜想是现代数学中一种深刻的统一力。虽然很大程度上是开放的，但即使是目前已知的，也会对数论、表示理论、代数几何甚至数学物理产生影响。研究的基本对象是自同构表征。自同构表示是使用局部对称空间来定义的，局部对称空间是高度对称的空间，可以被认为是任意维度的鼓。鼓上的每一个声音都是纯音的叠加，自同构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，而功能猜想预测，当人们可以关联l群时，就可以关联（更复杂的）自同构表示。PI还将在这一研究领域培训研究生，并参与从高中到博士后的各级指导活动。利用l群之间的关系可以定义称为l函数的函数，证明l函数是“好的”就足以证明朗兰泛函性。这个项目将开发新的技术来证明l函数是好的。用来证明l函数是好的最古老的工具是从傅里叶分析、信号处理等中熟悉的泊松求和公式。PI和B. Liu最近首次发现了泊松求和公式的推广。研究的更具体的目标是扩展和利用这些推广，目的是研究高阶的三重积l函数。利用由Cogdell和Piatetski-Shapiro提出的逆理论，许多朗兰兹泛函可以简化为对三重积l函数的研究，因此这是非常重要的。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.