Explicit Local Langlands Correspondences

局部Langlands对应是两种对称性之间的相互作用：复李群的结构和局部伽罗瓦理论。本提案中的工作将建立显式局部Langlands猜想的新情况，特别强调E8这样的群，它们源于二十面体对称。提议者将在他以前的工作的基础上，首先完成一个正在进行的项目，旨在明确验证驯服的朗兰兹对应关系，其次通过扩展范围来验证某些情况下的野生对应关系。 对称性是数学中的一个基本概念。表示论是研究对称性如何以不同的方式表现出来的。这个建议涉及两种完全不同的对称性的相互作用，其中一种是几何的，如在二十面体中，另一种是代数的，由多项式根之间的对称性引起。这种对称性的相互作用是数学发展的一个分支，其根源可以追溯到欧几里得的《几何原本》。这一提议的更广泛影响的一部分是通过使用古代数学的共同语言来统一不同的数学社区，来认识和分享当地朗兰兹对应的历史组成部分。例如，并不是每个人都能理解E8，但它源于欧几里得，它可以被广泛研究，而且一度被广泛研究，因为各种各样的数学之外的原因，这些原因对所有受过教育的公民都很重要。

The Local Langlands correspondence is a conjectural interaction between two kinds of symmetry: the structure of complex Lie groups and local Galois theory.The work in this proposal will establish new cases of the explicit local Langlands conjecture, with special emphasis on groups such as E8, which arise from icosahedral symmetry. The proposer will build on his previous work in this direction by first completing an ongoing project aimed at explicitly verifying the tame Langlands correspondence and second by extending the scope to verify the wild correspondence in some cases. Symmetry is a fundamental concept in Mathematics.Representation Theory is the study of how symmetry manifests in different ways. This proposal concerns the interactions of two quite different kinds of symmetry, one of them geometric, as in the icosahedron, and the other algebraic, arising from the symmetries among roots of polynomials. This interaction of symmetries is a branch of mathematical development whose roots go back to Euclid's Elements. Part of the Broader Impact of this proposal is to recognize and share this historical component of the local Langlands correspondence, by using the common language of ancient mathematics to unify disparate mathematical communities. For example, not everyone can understand E8, but it has roots in Euclid, which can be, and at one time was studied by a wide population, for diverse extra-mathematical reasons that are important for all educated citizens.