Geometric Methods in Representation Theory and the Langlands Program

表征理论是用线性代数来研究对称性的。因此，它与广泛的数学领域密切相关，并具有丰富的应用，例如粒子物理，物理化学和计算机视觉。自近一个世纪前这门学科出现以来，表征理论与代数几何（研究多项式方程的解）之间的相互作用影响了每个领域的景观。最近，在过去的几十年里，数论在这个世界上扮演着越来越重要的角色。朗兰兹程序是一个影响深远的猜想网络，它预测了深度数论和表示论现象之间意想不到的匹配。这个项目包括学生研究的机会。这个项目的中心是在朗兰兹计划的框架内表示理论的几何方法。长期目标是建立p进群表示的几何理论。这将允许在p进群表示的传统代数方法的快速发展和朗兰兹对应的几何进展的快速发展之间进行交流。此外，它将l包的猜想代数构造与深层几何现象联系起来。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。

Representation theory is the study of symmetries using linear algebra. As such, it is deeply intertwined with a vast range of mathematical fields and has rich applications, to particle physics, physical chemistry, and computer vision for example. Since the dawn of the subject nearly a century ago, interactions between representation theory and algebraic geometry, which is the study of solutions of polynomial equations, have influenced the landscape of each area. More recently, in the past several decades, number theory has played an increasing important role in this world. The Langlands program is a far-reaching web of conjectures which predicts unexpected matchings between deep number-theoretic and representation-theoretic phenomena. This project includes opportunities for student research. This project is centered on geometric methods in representation theory within the framework of the Langlands program. The long-term objective is to construct a geometric theory of representations of p-adic groups. This will allow passage between rapid developments in the traditionally algebraic approach to representations of p-adic groups and rapid developments in geometric progress in the Langlands correspondence. Moreover, it will relate conjectural algebraic constructions of L-packets to deep geometric phenomena.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.