Derived and perverse methods in the local Langlands correspondence

朗兰兹纲领是一系列影响深远的理论，可以追溯到20世纪70年代，它涉及到数论、表示论、调和分析和数学物理等广泛分离的数学分支。朗兰兹纲领的核心是局部朗兰兹对应，它将p进群的可接受表示（调和分析的对象）与伽罗瓦表示（数论的对象）联系起来。最近人们已经清楚地认识到，这种对应关系两边的物体都有非常丰富的几何结构，而局部朗兰兹对应关系常常允许我们研究对应关系一边的几何结构，从而推断出另一边的几何结构。一个自然的问题是：一边的几何结构在多大程度上反映在另一边的几何结构中？最好的要求是用相应的伽罗瓦表示来描述p-adic群的容许表示范畴。最近许多人的突破表明，这不仅是合理的，但在特殊情况下可能是触手可及的。本研究的目标是：第一，获得这样的描述在设置，它是可行的，这样做，第二，探索这样的描述的后果，各种应用的局部朗兰兹对应在数论。

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.