Representations of p-adic Groups and the Local Langlands Correspondence

该项目是由朗兰兹计划推动的。朗兰兹计划是一个影响深远的猜想集合，涉及不同的数学领域，包括数论和表示论，因此提供了一个强大的工具，通过在不同领域之间转移结果来证明定理和寻找解释。朗兰兹计划最近的工作导致了各种主要猜想的解决。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成的对象，例如方程的整数解。除了它在数学中的基础作用外，数论还对我们的日常生活产生了重要影响，例如，通过密码学。另一方面，表示理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，人们可以将抽象问题归结为线性代数中的问题，这通常更容易学习。表象理论在数学之外也有应用，例如，在物理学中。这个项目将加深我们对表示论方面的研究对象的理解，并旨在取得进一步的进展，在数论方面显式地构造局部朗兰兹对应。p-Add群的表示理论中的一个中心开放问题是构造所有光滑的、复杂的、超极小的表示。在这个项目中，PI将提供一种新的这种表示的构造，并证明它对所有素数p都是穷举的。同时，PI和她的合作者将探索另一种新的方法来更好地了解p-Add群的表示的结构。此外，PI打算利用新的陈述结构来获得关于当地朗兰兹通信的新见解。这一奖项反映了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 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.