TITLE: Tools for the Langlands Program?
QUESTION [35 upvotes]: Hi,
I know this might be a bit vague, but I was wondering what are the hypothetical tools necessary to solve the Langlands conjectures (the original statments or the "geometic" analogue). What I mean by this is the following: for the Weil Conjectures it became clear that, in order to prove them, one needed to develop a marvelous cohomology theory that would explain Weil's observations.  Of course, we all know that etale cohomology is that marvelous tool.  By analogy, what "black box" tools are necessary for the Langlands program? Broadly speaking what tools do we need for the Langlands program?
Curious grad student,
Ben

REPLY [7 votes]: I don't know a whole lot about the Langlands program, but if there is one tool that seems to come up a lot in geometric Langlands, it's perverse sheaves.  You see a lot of singular algebraic varieties in geometric Langlands, and perverse sheaves are meant as a singular generalization of a vector bundle with a flat connection.  Ordinary sheaves are already a singular generalization of vector bundles, but not the relevant one.  Perverse sheaves (which are made from sheaves but not sheaves themselves) are a more apropos generalization that incorporates and sort-of just is intersection (co)homology.
I can also say that I wasn't going to learn about perverse sheaves until I had to.  However, I have now seen several important papers, in the related categorification program, that read this way:  "Perverse sheaves + necessary restrictions = a good solution".  So now I might be slowly getting used to them.  I can also see that even the formalism perverse sheaves or intersection homology is sort-of inevitable.  In some of the simpler constructions, the varieties (over $\mathbb{C}$, say) are non-singular and certain answers arise as ordinary cohomology products or intersection products.  For instance, the Schubert calculus in a Grassmannian manifold.  What choice do you have if the Grassmannian is replaced by a singular variety $X$?  For some of these categorification/Langlands questions, you can either propose wrong answers, or ad hoc answers, or you can automatically get the right answer by using intersection homology on $X$.  (With middle perversity, as they say.)