** The short course will be leaded by Professor Cheng-Chiang Tsai (Stanford University).

The goal of this short course is to give a quick introduction to perverse sheaves, intersection cohomology, and their application to algebraic geometry. On one hand, perverse sheaves provide a replacement of the singular cohomology for non-smooth algebraic varieties and morphisms, that in some occasions works as if they were smooth. On the other hand, perverse sheaves were first discovered for linear differential equations with regular singularities, a setting which is connected to ours via the Riemann-Hilbert correspondence.

We will mostly work with algebraic varieties over the complex numbers and the analytic topology on them. We begin with a review on Grothendieck’s six functors. After that we work on the triangulated category of complexes of constructible sheaves, t-structures, and eventually the abelian subcategory of perverse sheaves and their construction from intersection cohomology sheaves.

The most important result about perverse sheaves is the decomposition theorem. In spirit this is the result that allows us to sometimes work with proper morphisms of varieties as if they are also smooth morphisms, or rather to measure the failure of smoothness for the sake of cohomology. The original proof of the decomposition theorem by Beilinson, Bernstein, Deligne and Gabber uses the theory of weight in Deligne’s second proof of the Weil conjectures.

For the sake of time we will necessarily skip the detail of some proofs, which likely include the proof that the six functors preserve constructible sheaves and the proof of the decomposition theorem.

As an application, we will work out the Betti numbers of the Hilbert scheme (a compactified moduli space) of points on a smooth surface. If time permits, we will discuss other applications, possibly on topics in geometric representation theory and toric varieties.