The aim of the course is to introduce the theory of perverse sheaves, which provides a powerful tool in understanding the topology of algebraic varieties starting from the behavior of maps between them.

During the lectures, I will introduce the key ideas of the theory. While it is note in the scope of the course to give detailed proofs of results, the focus will be given to exercises and examples. 

Below is a list of the topics, which will be treated: 

1. Review of the Hodge-Lefschetz package for smooth projective varieties 

2. Complexes of sheaves, review of derived categories of coherent and constructible sheaves, together with their natural functors

3. Direct images, Leray-Hirsch theorem

4. Intersection cohomology: topoligical and sheaf theoretic definition

5. Perverse sheaves and decomposition theorem

6. Applications of the theory

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REFERENCES: 

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• M. De Cataldo, Lectures on perverse sheaves and decomposition theorem, available at https://www.math.stonybrook.edu/~mde/AnnArborLectures.pdf

• De Cataldo, L. Migliorini, The decomposition theorem, perverse sheaves and the topology of algebraic maps, Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 4,535–633.

• M. Goresky, R. Macpherson, Intersection homology theory, Topology 19 (1980), no. 2, 135–162.

• M. Goresky, R. Macpherson, Intersection homology, II. Invent. Math. 72 (1983), no. 1, 77–129.

• M. Laurentiu, Intersection cohomology and perverse sheaves with applications to singularity (book)

• G. Williamson, An illustrated guide to perverse sheaves, available at http://people.mpim-bonn.mpg.de/geordie/perverse_course/lectures.pdf

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