Functoriality for Relative Trace Formulas

朗兰兹函数性猜想（英语：Langlands functoriality consumption），即“不同的算术鼓共享一些共同的本征频率”，在数论中有着巨大的应用，其中包括拉马努金和其他人关于称为自守形式的特殊函数的系数大小的百年理论。PI和他的合作者已经将这个猜想扩展到所谓的“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，例如在最近的Gan，Gross和Prasad的著作中。到目前为止，证明函性的重要例子的主要工具是迹公式，但在目前的形式下，它几乎达到了极限。这个项目将研究如何证明这些命题的使用量子化的想法，其起源在于数学物理。这个想法将用于构建比较（相对）迹公式的新方法，大大扩展其潜在范围和适用性。该项目的更广泛的影响包括会议组织和指导研究生。PI已经表明，在以前的工作中，在一些低秩的情况下，可以通过相对迹公式之间的一些新的“转移算子”来建立相对函子性。这种非标准的迹线公式比较在朗兰兹的“超越内窥镜”建议中被设想;“相对”设置允许更大的灵活性，以及更多的潜在应用，以探索这种比较。以前的工作主要集中在与相对迹公式相关的L-群秩为1的情况。这个项目的主要目标是研究如何将迁移算子的构造推广到更高的秩。其主要思想是将迹公式视为其余切堆栈的量子化，而余切堆栈又在很大程度上由L群控制。利用这些余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。在另一个单独的轨道上，该项目将继续研究PI最近与Ben-Zvi和Venkatesh的工作中所揭示的Hamilton空间的对偶性，目的是将这种对偶性扩展到超球环境之外，并探索p-的表示理论的应用，该奖项反映了NSF的法定使命，并被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.