• Reading Ginzburg's paper
• Writing my own paper

So, I got curious about Geometric Langlands. I have heard about the two words, and also used it myself, but I don't know exactly what I am talking about. There are some equivalence of categories, which is a nice way to package my ignorance. There is a hard side, and an easy side. The hard side involves certain constructible sheaves on $Bun_G$ with some singular support in the global Nilpotent cone, the easy side involves some local system on the same curve but valued in the dual Langlands group.

Is the abelian case known?

What is an example of this? Some sample curve, and sample group?

Under some nice cases, the local Hecke eigensheaf in $D^bSh(Bun_G(X))$ is also a global Hecke eigensheaf. How to think about $Bun_G$ and sheaves on it? A point in $Bun_G$ is a $G$-bundle over $X$, but we are not just thinking about one G-bundle, and we are not moving one G-bundle to another. So, what is Hecke transformation?

Well, this is just like matrix and vectors. We use vector to store numbers on a bunch of points, and we use matrix to recombine them. If we replace the finite set of points by some 'continuous space', then what we do is integral transform on space of functions (or distributions) on that space. Now, we categorify. Still over a finite collection of points, we can consider a bunch of vector spaces. We want to output a new set of vector spaces. How to build a new vector space?

• there is something called pushforward. What does that mean? It is like taking limit of some diagram (does taking projective limit commute with taking $Hom(X, -)$? limit is the universal receiver of Hom-to). For example, taking intersection of two subspaces is taking 'limit'.