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

数论是研究整数和由它们构建的对象的学科。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，而表示论是使用矩阵研究群。虽然数论和表示论是非常不同的数学领域，但朗兰兹对应预言了它们之间迷人的联系。围绕朗兰兹对应的思想是数学中许多突破性进展的驱动力，包括著名的费马大定理证明，这是一个经受了数学家近400年努力的猜想。这个项目涉及解决一个长期存在的问题，关于朗兰兹对应的表示论方面，并研究数论的桥梁。PI还将组织研讨会和活动，以帮助未来的数学家在他们的职业发展和支持代表性不足的群体。局部朗兰兹对应的表示论方面的一个基本问题是所有p-adic群的所有超尖点表示的构造，这是该项目的第一个主要目标。解决这个问题将涉及处理与驯服情况相比在非驯服情况下出现的所有并发症。基于这些结果，PI还将推进Langlands计划的各个方面：通过构造自守形式之间的同余的整体Langlands程序，通过证明有限重性结果的相关Langlands程序，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。

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.