Geometric Methods in Representation Theory and the Langlands Program

表示论是用线性代数研究对称性的学科。因此，它与广泛的数学领域深深交织在一起，并具有丰富的应用，例如粒子物理学，物理化学和计算机视觉。自近世纪前这一学科出现以来，表示论与代数几何（即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，数论在这个世界上发挥了越来越重要的作用。朗兰兹纲领是一个影响深远的网络，它预测了深层数论和表示论现象之间意想不到的匹配。该项目包括学生研究的机会。这个项目的重点是几何方法在朗兰兹计划的框架内表示理论。长期目标是建立一个p进群的几何表示理论。这将允许通过之间的快速发展，在传统的代数方法表示的p-adic组和快速发展的几何进步的朗兰兹对应。此外，它将涉及的L-包的代数结构，以深几何现象。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.