Derived and perverse methods in the local Langlands correspondence

朗兰兹计划是一系列影响深远的猜想，可以追溯到20世纪70年代，这些猜想将数论、表示论、调和分析和数学物理等广泛独立的数学分支联系在一起。朗兰兹程序的核心是局部朗兰兹对应，它将p-进群的可接受表示(调和分析对象)与伽罗瓦表示(数论对象)联系起来。最近清楚的是，这种通信双方的对象都具有非常丰富的几何结构，并且局部朗兰兹通信通常允许我们研究通信的一侧的几何来推导关于另一侧的几何的结论。一个自然的问题是：准确地说，一侧的几何在多大程度上反映在另一侧的几何中？最优的要求是用相应的伽罗瓦表示来描述p-进群的可容许表示的范畴。最近许多人的突破表明，这不仅是可能的，而且在特殊情况下可能是触手可及的。本研究的目标是：第一，在可行的环境中获得这样的描述，第二，探索这样的描述在数论中局部朗兰兹对应的各种应用中的后果。

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 objects on both sides of this correspondence have a very rich geometric structure, and that the local Langlands correspondence often allows us to study the geometry on one side of the correspondence to deduce conclusions about the geometry on the other side.A natural question to ask is: to precisely what extent is the geometry of one side reflected in the geometry of the other? The most optmistic thing to ask for would be a description of the category of admissible representations a p-adic groups in terms of the corresponding Galois representations. Recent breakthroughs by many people suggest that this is not only plausible, but might be within reach in special cases.The goals of this research are: first, to obtain such descriptions in settings where it is feasible to do so, and second, to explore the consequences of such a description for various applications of the local Langlands correspondence in number theory.