The Gan-Gross-Prasad Conjecture: Archimedean Theory

该奖项支持首席研究员在数论和表象理论方面的研究。数论起源于对自然数的研究。它是最古老的数学分支之一。在过去的半个世纪里，它已经成为一种不可或缺的工具，在数据传输和处理、通信系统和网络安全等领域有着广泛的应用。表象理论是在人们试图解代数方程的过程中产生的。这门学科成熟于20世纪，至今仍在快速发展。它研究物体的对称性，几乎应用于数学、物理学、宇宙学和材料科学的所有分支。这个项目将探索这两个领域的一些主要猜测。此外，该奖项还将支持该协会所在机构的研究生。20世纪60年代，朗兰兹提出了一系列关于表象理论和数论的深刻猜想，这些猜想从那时起就成为数论家和表象理论家的指导原则。20世纪90年代，在朗兰兹哲学的指导下，Gross和Prasad提出了几个关于正交群表示理论的猜想。与Gan一起，他们将这些猜想扩展到包括所有经典群。这些猜想及其改进和算术对应的猜想站在数论、表示论和算术几何的十字路口。该项目的目标是研究这些猜想。更详细地，我们研究了以下两个问题：(1)实正交群的局部Gan-Gross-Prasad猜想；(2)由Gan-Gross-Prasad猜想的算术版本产生的算术基本引理的阿基米德对应。对于第一个问题，我们应用了theta对应的方法，以及Shahimi局部系数的一个相对版本，这是我们的主要创新点。这种方法还给出了正交群的L包的一个归纳构造，这些正交群本身就是朗兰兹猜想中的重要研究对象。对于第二个问题，我们利用了相应的轨道积分和球面特征的归纳结构。它使用Zeta积分加倍的理论将问题简化为显式计算。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.