Representations of p-adic Groups and the Local Langlands Correspondence

这个项目的动机是朗兰兹计划。朗兰兹程序是一个意义深远的集合，它涉及数学的不同领域，包括数论和表示论，从而提供了一个强大的工具来证明定理，并通过在不同领域之间转移结果来找到解释。最近在朗兰兹纲领中的工作导致了各种主要问题的解决。数论是纯数学中最古老的领域之一，它包括研究整数和由它们构建的对象，例如方程的整数解。除了它在数学中的基本作用外，数论对我们的日常生活也有着重要的影响，例如通过密码学。另一方面，表示论是使用矩阵和线性代数来研究抽象对象，例如立方体的对称性（保持立方体的反射和旋转）。通过将抽象对象实现为矩阵，可以将抽象问题简化为线性代数中的问题，这些问题通常更容易研究。表示论在数学之外也有应用，例如在物理学中。这个项目将提高我们的理解的对象研究的表示论方面，并旨在实现进一步的进展对一个明确的建设当地朗兰兹对应的数论方面。一个中心的开放问题表示理论的p-adic群是建设所有顺利，复杂，supercupsidal表示。在这个项目中，PI将提供一个新的结构，这样的表示，并证明其详尽的所有素数p平行，PI和她的合作者将探索另一种新的方法，以获得更好地理解的结构表示的p-adic组。此外，PI打算使用新的表述结构来获得有关当地朗兰兹通信的新见解。该奖项反映了NSF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。

This project is motivated by the Langlands program. The Langlands program is a far-reaching collection of conjectures that relates different areas of mathematics, including number theory and representation theory, and thereby provides a powerful tool to prove theorems and find explanations by transferring results between different areas. Recent work in Langlands program has lead to the resolution of various major conjectures. Number theory is one of the oldest areas in pure mathematics, which includes the study of integers and objects built out of them, such as integer solutions to equations. Apart from its fundamental role inside mathematics, number theory manifests important influences on our everyday life, for example, through cryptography. Representation theory, on the other hand, is the study of abstract objects, such as the symmetries of a cube (reflections and rotations that preserve the cube), using matrices and linear algebra. By realizing abstract objects as matrices, one can reduce abstract problems to questions in linear algebra, which are often easier to study. Representation theory has also applications outside of mathematics, for example, in physics. This project will enhance our understanding of the objects studied on the representation theory side and aims to achieve further progress towards an explicit construction of the local Langlands correspondence on the number theory side.One of the central open questions in the representation theory of p-adic groups is the construction of all smooth, complex, supercupsidal representations. In this project, the PI will provide a new construction of such representations and prove its exhaustiveness for all primes p. In parallel, the PI and her collaborators will explore another novel approach to obtain a better understanding of the structure of the representations of p-adic groups. Moreover, the PI intends to use the new construction of representations to gain new insights about the local Langlands correspondence.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.