Restriction Problems in Representation Theory

这个项目的一个中心主题是朗兰兹计划，它令人惊讶地融合了群分析和数论。一个基本的例子是圆上的函数作为三角函数之和的傅里叶展开。圆圈是一个非常对称的物体。我们可以旋转它的任何角度，而不改变它的形状。圆的旋转形成了数学上所说的一组变换，或简称为一组。圆上的每个函数都有一个傅里叶展开，在傅里叶展开式中，它被写成三角函数的和。三角函数是圆上最简单的函数，因此对圆上的函数或周期函数的分析就简化为三角函数。一个多世纪以来，人们就知道傅里叶展开式及其推广承载了大量的数论信息。在这个例子的激励下，在过去的50年里，一大类问题被统一到一个被称为朗兰兹计划的方案中。粗略地说，目标是将组上的任何函数分解为最简单的可能函数的和。这个分解问题是这个项目的主旋律。该项目将包括对研究生的培训。更具体地说，该项目的主要目标是完成对G2组的朗兰兹分类。这个群是由尤金·迪克森发现的，它被发现在物理学和弦理论中扮演着重要的角色。本研究使用的主要工具是PI在以前的工作中发展的例外题元对应理论。此外，还将把特殊的theta对应应用于对偶实群和p-ady群的超尖表示。该项目将包括对研究生的培训。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。

A central theme in this project is Langlands program, which is a surprising mixture of analysis on groups and number theory. A basic example is the Fourier expansion of a function on the circle as a sum of trigonometric functions. The circle is a very symmetric object. We can rotate it for any angle without changing its shape. Rotations of a circle form what is called in mathematics a group of transformations, or simply a group. Every function on the circle has a Fourier expansion in which it is written as a sum of trigonometric functions. Trigonometric functions are the simplest functions on the circle, and so the analysis of functions on the circle, or periodic functions, is reduced to trigonometric functions. It has been known for over a century that Fourier expansions and their generalizations carry significant number theoretic information. Motivated by this example, over the past fifty years a large class of problems has been unified in a scheme known as the Langlands program. Roughly speaking, the goal is to decompose any function on a group as a sum of the simplest possible functions. This decomposition problem is the main theme of this project. This project will include training of graduate students.More specifically the main goal of this project is to complete the Langlands classification for the group G2. This group was discovered by Eugene Dickson and it has been found to play an important role in physics and string theory. The principal tool employed in this research is the theory of exceptional theta correspondences that was developed by the PI in previous work. Applications of the exceptional theta correspondence to dual pairs of real groups and to supercuspidal representations of p-adic groups will developed as well. This project will include training of graduate students.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.