Derived and perverse methods in the local Langlands correspondence

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，在这个对应的两边的物体都有非常丰富的几何结构，而且局部朗兰兹对应通常允许我们研究对应的一边的几何来推断另一边的几何结论。一个很自然要问的问题是：在多大程度上，一边的几何形状反映在另一边的几何形状上？最乐观的要求是用相应的伽罗瓦表示来描述p进群的可容许表征范畴。许多人最近的突破表明，这不仅是合理的，而且在特殊情况下可能是可以实现的。本研究的目标是：首先，在可行的情况下获得这样的描述，其次，探索这种描述对数论中局部朗兰兹对应的各种应用的影响。

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.