The Gan-Gross-Prasad Conjecture: Archimedean Theory

该奖项支持首席研究员对数论和表示论的研究。数论的历史根源在于对自然数的研究。它是最古老的数学分支之一。在过去的半个世纪，它已成为一个不可或缺的工具，在数据传输和处理，通信系统和互联网安全等领域的各种应用。表示论起源于人们求解代数方程的尝试。这门学科在世纪发展成熟，至今仍在快速发展。它研究物体的对称性，并应用于数学、物理学、宇宙学和材料科学的几乎所有分支。本项目将探讨这两个领域的一些主要问题。此外，该奖项将支持PI机构的研究生。在20世纪60年代，朗兰兹制定了一系列关于表示论和数论的深刻的理论，这些理论从那时起就成为数论和表示论的指导原则。在20世纪90年代，格罗斯和普拉萨德在朗兰兹的哲学指导下，制定了几个关于正交群表示论的理论。与甘一起，他们将这些理论扩展到包括所有经典群。这些几何学及其精化和算术对应物处于数论、表示论和算术几何的十字路口。该项目的目标是研究这些结构。更详细地，我们研究了以下两个问题：（1）真实的正交群的局部Gan-Gross-Prasad猜想;（2）Gan-Gross-Prasad猜想的算术版本所产生的算术基本引理的阿基米德对应。对于第一个问题，我们采用的方法θ对应，和一个相对版本的Shahidi的本地系数，这是我们的主要创新点。这种方法也给出了正交群的L-包的一个归纳构造，正交群本身就是朗兰兹定理中的重要研究对象。对于第二个问题，我们利用相关的轨道积分和球面字符的归纳结构。该奖项反映了NSF的法定使命，并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，被认为值得支持。

This award supports the principal investigator's research on Number Theory and Representation Theory. Number theory has its historical roots in the study of natural numbers. It is among the oldest branches of mathematics. Within the last half century it has become an indispensable tool, with diverse applications in areas such as data transmission and processing, communication systems, and internet security. Representation Theory emerges from people’s attempt to solve algebraic equations. The subject matured in the 20th century and is still fast developing. It studies the symmetries of objects and has applications to almost all branches of mathematics, physics, cosmology and material science. This project will explore some major conjectures in these two areas. Additionally, this award will support graduate students in the PI's institution. In the 1960s Langlands formulated a series of profound conjectures relating representation theory and number theory, and these conjectures have become guiding principles for number theorists and representation theorists since then. In the 1990s, Gross and Prasad formulated several conjectures on representation theory of the orthogonal groups, guided by the philosophy of Langlands. Together with Gan, they extended these conjectures to include all classical groups. These conjectures and their refinement and arithmetic counterparts stand at the crossroads of number theory, representation theory, and arithmetic geometry. The goal of the project is to study these conjectures. In more detail, we study the following two problems (1) the local Gan--Gross--Prasad conjecture for real orthogonal groups; (2) the archimedean counterpart of the arithmetic fundamental lemma arising from an arithmetic version of the Gan--Gross—Prasad conjecture. For the first problem we apply the method of theta correspondences, and a relative version of Shahidi’s local coefficients which is our main point of innovation. This method also gives an inductive construction of L-packets for orthogonal groups, which are themselves important objects of study in the conjectures of Langlands. For the second problem, we make use of an inductive structure of the relevant orbital integrals and spherical characters. It reduces the problem to explicit calculations using the theory of doubling zeta integrals.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.