Functoriality for Relative Trace Formulas

朗兰兹泛函猜想，即“不同的算术鼓有一些共同的特征频率”，在数论中有着巨大的应用，其中包括由拉马努金和其他人提出的关于称为自同构形式的特殊函数的系数大小的百年猜想。PI和他的合作者将这一猜想扩展到所谓的“相对”设置，其中包括研究l函数（也称为zeta函数）的特殊值的方法，例如Gan， Gross和Prasad的突出和最近的猜想。迄今为止，证明重要功能实例的主要工具一直是跟踪公式，但以其目前的形式，它几乎已经达到了极限。这个项目将通过使用量子化的概念来检验证明这些猜想的方法，量子化的概念起源于数学物理学。这一思想将用于构建比较（相对）迹公式的新方法，大大扩大其潜在的范围和适用性。该项目更广泛的影响包括组织会议和指导研究生。PI在先前的工作中已经表明，在一些低秩的情况下，人们可以通过一些新的“转移算子”在相对的迹公式之间建立相对的泛函性。朗兰兹的“超越内窥镜检查”提案设想了这种非标准的痕量公式比较；“相对”设置为探索这种比较提供了更大的灵活性和更多潜在的应用。以往的工作主要集中在与相对示踪公式相关联的l群为一级的情况下。这个项目的主要目标是研究将转移算子的构造推广到更高阶的方法。主要思想是将迹公式视为其余切堆栈的量化，而余切堆栈又在很大程度上由l群控制。利用这些共切堆栈之间的自然对应关系，该项目旨在构建它们的量化之间的转移算子。在一个单独的轨道上，该项目将继续研究在PI最近与Ben-Zvi和Venkatesh的工作中推测的哈密顿空间的对偶性，目的是将这种对偶性扩展到超球面设置之外，并探索p进群的表示理论的应用。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 Langlands functoriality conjecture, that "different arithmetic drums share some common eigenfrequencies," has immense applications in number theory, among others to the century-old conjectures due to Ramanujan and others about the size of coefficients of special functions called automorphic forms. The PI and his collaborators have broadened this conjecture to the so-called "relative" setting, which includes methods of studying special values of L-functions (also called zeta functions), such as in the prominent, and more recent, conjectures of Gan, Gross, and Prasad. The main tool for proving important instances of functoriality so far has been the trace formula, but in its current form it has nearly reached its limits. This project will examine ways to prove these conjectures by use of the idea of quantization, whose origins lie in mathematical physics. This idea will be used to construct novel ways of comparing (relative) trace formulas, drastically expanding their potential reach and applicability. The broader impacts of the project include conference organization and mentoring of graduate students.The PI has already shown, in prior work, that in some low-rank cases one can establish relative functoriality via some novel "transfer operators" between relative trace formulas. Such non-standard comparisons of trace formulas were envisioned in Langlands's "Beyond Endoscopy" proposal; the "relative" setting allows for more flexibility, and more potential applications, for the exploration of such comparisons. Prior work was focused mostly on the case when the L-groups associated to the relative trace formulas are of rank one. The main goal of this project will be to examine ways to generalize the construction of transfer operators to higher rank. The main idea is to view a trace formula as the quantization of its cotangent stack, which in turn is largely controlled by the L-group. Using natural correspondences between such cotangent stacks, the project aims to construct transfer operators between their quantizations. On a separate track, the project will continue work on the duality of Hamiltonian spaces conjectured in the PI's recent work with Ben-Zvi and Venkatesh, with the aim of extending this duality beyond the hyperspherical setting, and exploring applications for the representation theory of p-adic groups.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.