CAREER: Representations of p-adic groups and different incarnations of the Langlands Program

数论是对整数和由它们构成的对象的研究。群是对对称性进行编码的抽象数学对象，例如立方体或晶体的对称性，而表示理论是使用矩阵研究群的。虽然数论和表示论是非常不同的数学领域，但朗兰兹的通信预言了它们之间令人着迷的联系。围绕朗兰兹通信的思想是数学中许多开创性进步的驱动力，包括著名的费马最后定理的证明，这一猜想经受住了数学家近400年的努力。这个项目涉及解决朗兰兹通信中关于表示理论方面的一个长期存在的问题，并研究通向数论的桥梁。国际数学家联合会还将组织讲习班和活动，以协助有前途的数学家的职业发展，并支持代表性不足的群体。在局部朗兰兹对应的表示理论方面的一个基本问题是构造所有p-Add群的所有超尖球面表示，这是本项目的第一个主要目标。解决这一问题将涉及处理与温顺情况相比在非驯服情况下出现的所有复杂情况。基于这些结果，PI还将推进朗兰兹计划的各个方面：通过构建自同构形式之间的同余来推进全球朗兰兹计划，通过证明有限重数结果来证明相对朗兰兹计划，以及显式的局部朗兰兹对应。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。

Number theory is the study of the integers and objects built out of them. Groups are abstract mathematical objects which encode symmetries, for example the symmetries of a cube or crystal, and representation theory is the study of groups using matrices. Although number theory and representation theory are very different areas of mathematics, the Langlands correspondence predicts a fascinating connection between them. The ideas surrounding the Langlands correspondence are the driving force for many groundbreaking advances in mathematics including the famous proof of Fermat's Last Theorem, a conjecture that had withstood mathematicians' effort for almost 400 years. This project involves tackling a long-standing question concerning the representation theory side of the Langlands correspondence and studying the bridge to number theory. The PI will also organize workshops and activities to assist up-and-coming mathematicians in their career development and support underrepresented groups. A fundamental problem on the representation theory side of the local Langlands correspondence is the construction of all supercuspidal representations for all p-adic groups, which is the first main objective of this project. Solving this problem will involve tackling all the complications that arise in the non-tame case compared to the tame case. Based on these results, the PI will also advance various aspects of the Langlands program: the global Langlands program by constructing congruences between automorphic forms, the relative Langlands program by proving finite multiplicity results, and the explicit 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.