The local Langlands correspondence in families

朗兰兹纲领是一系列影响深远的猜想，可以追溯到20世纪70年代，它涉及数论、表示理论、谐波分析和数学物理等广泛独立的数学分支。朗兰兹纲领的核心是局部朗兰兹对应，它将p进群的可容许表示（调和分析的对象）与伽罗瓦表示（数论的对象）联系起来。最近已经很清楚，为了正确理解这种对应关系，我们不应该只考虑对应关系两边的单个对象，而是考虑这种对象的整个家族，它们随着某些参数的变化而变化。这自然导致了以下问题：如果一个人改变对应的一边的对象，对应的另一边的对象如何变化？拟议的研究旨在开发一个解决这个问题的本地朗兰兹对应的版本。此外，这个问题的答案应该在朗兰兹程序中有广泛的应用，特别是对埃默顿的“p-adic全球朗兰兹对应”。我们试图探索这些应用，并通过这些技术在朗兰兹程序中推断出新的结果。

The Langlands program is a series of far-reaching conjectures, dating back to the 1970s, that relate such widely separate branches of mathematics as number theory, representation theory, harmonic analysis, and mathematical physics. Central to the Langlands program is the local Langlands correspondence, which relates admissible representations of p-adic groups (objects from harmonic analysis)to Galois representations (objects from number theory). It has recently become clear that in order to properly understand this correspondence, we should consider not just single examples of objects on both sides of the correspondence, but whole families of such objects, that vary as one varies certain parameters.This leads naturally to the following question: if one varies the objects on one side of the correspondence, how do the objects on the other side of the correspondence change? The proposed research seeks to develop a version of the local Langlands correspondence that addresses this question. Moreover, the answer to this questions should have extensive applications within the Langlands program, particularly to Emerton's "p-adic global Langlands correspondence". We seek to explore these applications and deduce new results in the Langlands program via these techniques.