Sheaves are mathematical objects which are useful in answering the types of topological questions that usually come up in algebraic geometry. For example, suppose we have a family of varieties. We might ask how the cohomology groups of the members of the family vary. Sheaves are almost essential for formulating the question precisely. And sometimes they can be used to answer it.
In algebraic geometry, families of algebraic varieties can degenerate in natural and interesting ways. One example to keep in mind is a family of elliptic curves degenerating to a nodal cubic. In that case, the cohomology groups of the members of the family change dramatically at the point where the degeneration occurs. So the dimensions of the cohomology sheaves can jump in very complicated ways. Vaguely speaking, the category of perverse sheaves is an abelian category which is has all the topological information contained in the category of sheaves, but which is somehow better adapted than the category of sheaves at dealing with cohomology groups of families of projective varieites. The main theorem concerning this is the Decomposition Theorem of Beilinson, Bernstein and Deligne which (assuming smoothness of the total space) breaks the cohomology groups of the family into a direct sum of pieces in a canonical and (sometimes) computable way.
This Decomposition Theorem along with some of the other properties of perverse sheaves has numerous applications to algebraic geometry, representation theory and combinatorics. Some of the applications (especially in representation theory) rely heavily on the relationship, known as the Riemann-Hilbert correspondence, between perverse sheaves and D-modules. (Despite its historical sounding name, this Riemann-Hilbert correspondence is a theorem of Kashiwara and, independently, Mebkhout from the early 1980s.)
This is a research level class on a fairly advanced topic. There are two consequences of this.
Concerning (1), I expect to write up problems. But I think it would be better if those are done collaboratively and then presented in class.
Concerning (2), the two main prerequisites that are going to be hard to cover completely are Poincare-Verdier duality and Etale Cohomology. These need to be black boxed to some extent, but it is important to be able to still use the black box to compute.
The main goal is the Decomposition Theorem. If time permits, it would also be good to explain the Beilinson-Bernstein proof of the Kazhdan-Lusztig conjecture. My hope is to follow roughly the plan of the texts listed above. So we will start off by discussing triangulated categories, the derived category and t-structures. Then we will define categories of perverse sheaves for various pervsities. (A perversity is a certain type of numerical sequence which gives rise to a t-structure on the derived category.) Using this we can give the formula Deligne wrote down for the IC sheaf (used to compute the Goresky-MacPherson intersection cohomology of a singular variety). The most important perversity in the theory is the so-called middle perversity. The category of perverse sheaves for the middle perversity has several surprisingly wonderful properties. (For example, it is artinian.) We will go through those properties. Then we will talk about the connection to the Weil conjectures and mixed perverse sheaves. This leads to the decomposition theorem. If we have time after this, we will read Bernstein's notes listed above.