Representations of p-adic Groups and the Local Langlands Correspondence

这个项目是由朗兰兹项目推动的。朗兰兹计划是一个意义深远的猜想集合，涉及数学的不同领域，包括数论和表示理论，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传递结果来寻找解释。朗兰兹纲领最近的工作已经解决了许多重大的猜想。数论是纯数学中最古老的领域之一，它包括对整数和由整数构成的对象的研究，比如方程的整数解。除了它在数学中的基础作用外，数论对我们的日常生活也有重要的影响，例如通过密码学。另一方面，表征理论是对抽象对象的研究，比如立方体的对称性（保持立方体的反射和旋转），使用矩阵和线性代数。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问题，这通常更容易研究。表征理论在数学之外也有应用，例如在物理学中。这个项目将增强我们对表征理论方面的研究对象的理解，并旨在在数论方面取得进一步的进展，以明确地构建局部朗兰兹对应。在p进群的表示理论中，一个中心的开放问题是所有光滑的、复杂的、超副的表示的构造。在这个项目中，PI将提供这种表示的新结构，并证明其对所有素数p的穷尽性。与此同时，PI和她的合作者将探索另一种新的方法来更好地理解p进群的表示结构。此外，PI打算使用表征的新结构来获得关于局部朗兰兹对应的新见解。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。

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.