Restriction Problems in Representation Theory

这个项目的中心主题是朗兰兹程序，它是群分析和数论的惊人混合。一个基本的例子是函数在圆上的傅里叶展开式作为三角函数的和。圆是一个非常对称的物体。我们可以旋转它任意角度而不改变它的形状。一个圆的旋转形成了数学上所谓的一组变换，或者简单地说是一个群。圆上的每个函数都有傅里叶展开式它被写成三角函数的和。三角函数是圆上最简单的函数，所以对圆上的函数或周期函数的分析，可以简化为三角函数。一个多世纪以来，人们已经知道傅里叶展开及其推广携带着重要的数论信息。在这个例子的激励下，在过去的五十年里，一大批问题被统一在一个被称为朗兰兹纲领的方案中。粗略地说，目标是将群上的任何函数分解为尽可能简单的函数的和。这个分解问题是这个项目的主题。这个项目将包括培养研究生。更具体地说，这个项目的主要目标是完成G2组的朗兰兹分类。这个群是由尤金·迪克森发现的，人们发现它在物理学和弦理论中起着重要作用。本研究中使用的主要工具是由PI在以前的工作中开发的异常θ对应理论。在实群的对偶和p进群的超尖表示中，也将发展异常对应的应用。这个项目将包括培养研究生。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。

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.