The local Langlands correspondence via endoscopy, geometry, type theory, and their interplay

这个项目的目的是研究局部朗兰兹对应的p-adic群使用的工具，类型理论，代数和刚性几何，谱理论。局部朗甘斯对应是两种对称性之间的一种神秘的几何关系--一种是由系数在真实的、复数或p-adic域中的矩阵群提供的几何关系，一种是由真实的或p-adic域的伽罗瓦群提供的算术关系。在过去的十年里，这一领域取得了巨大的进展，但该理论在很大程度上仍然是不成熟的。PI和他的合作者将试图通过使用类型理论来建立新的代数对应，以及阐明类型与Rapoport-Zink空间的上同调之间的联系，特别是在低秩酉群的情况下。现有的结构将使用内窥镜的理论进行更深入的研究。此外，PI和他的合作者将研究任意秩酉群的局部和全局Langlands对应，使用由亚瑟开发的迹公式技术。Langlands计划汇集了数学的几个领域，包括算术，几何，分析和对称。通过在它们之间建立深刻而令人惊讶的联系，它提供了回答有关数字算术的非常困难的问题的方法，最近费马大定理的证明就是例证。对数字算术的深入理解反过来又是许多最新技术发展的基础，包括纠错码、压缩、加密和安全通信。该项目旨在通过探索几种方法之间的联系以及利用最近才出现的技术来加深我们对朗兰兹纲领的理解。

The aim of this project is to investigate the local Langlands correspondence for p-adic groups using the tools of type-theory, algebraic and rigid geometry, and spectral theory. The local Langands correspondence is a mysterious conjectural relationship between two kinds of symmetries -- a geometric one, offered by groups of matrices with coefficients in real, complex, or p-adic fields, and an arithmetic one, offered by the Galois group of the real or a p-adic field. The last decade has seen a tremendous progress in this area, but the theory is still largely conjectural. The PI and his collaborators will attempt to establish new cases of the conjectural correspondence through the use of type-theory, as well as to elucidate the connection between types and the cohomology of Rapoport-Zink spaces, in particular in the case of low-rank unitary groups. The already existing constructions will be investigated more deeply using the theory of endoscopy. Furthermore, the PI and his collaborators will study both the local and the global Langlands correspondences for unitary groups of arbitrary rank, using the trace-formula techniques developed by Arthur.The Langlands program brings together several areas of mathematics, including arithmetic, geometry, analysis, and symmetry. By establishing deep and surprising ties between them, it provides the means for answering very hard questions about the arithmetic of numbers, as exemplified by the recent proof of Fermat's Last Theorem. A sophisticated understanding of the arithmetic of numbers is in turn fundamental to many recent technological developments, including error-correcting codes, compression, encryption and secure communication. This project aims at deepening our understanding of the Langlands program by exploring connections between several approaches to it, as well as by exploiting techniques which have only recently become available.