Course Description: The goal of the first part of this class will be to motivate and introduce perverse sheaves from the point of view of complex algebraic geometry. We will begin by studying some beautiful theorems about the cohomology of smooth varieties. We will then introduce sheaves and derived categories as a useful way for understanding and reinterpreting statements about cohomology. Armed and motivated by these tools, we will study intersection cohomology ("sheaves"), which will allow us to generalize the theorems about the cohomology of smooth varieties to singular ones. Finally, we will introduce perverse sheaves which provide a categorical home and generalization for intersection cohomology sheaves.

In the second part of the class we may explore one approach (due to de Cataldo and Migliorni) to proving some deep theorems about perverse sheaves. We will most likely conclude the class with a study of some applications of perverse sheaves to representation theory (for example, the Kazhdan-Lusztig conjectures, Springer theory and the geometric Satake theorem).

Unfortunately, I do not plan to discuss the amazing and powerful world of D-modules, reduction-to-positive-characteristic, or mixed Hodge modules.

Update: The class is now over! We didn't do much more than what is described above as the "first part of the class"... oh well. You can find the beautifully live-TeXed notes of Chao Li on his webpage. Many thanks to Chao Li for taking notes and to everyone who showed up to the class for their interest, enthusiasm and questions!

Texts: There is no required text for the course. There are many different references for perverse sheaves. However, there is no one reference that I will be following. As mentioned above, lecture notes can be found on Chao Li's webpage. Some of the references I used in preparing the class were:

• the books of Voisin and de Cataldo on Hodge theory
• Gel'fand-Manin's book "Homological Algebra"
• Kashiwara-Schapiro (mostly "Sheaves on Manifolds" but a little of "Sheaves and Categories")
• Dimca's book "Sheaves in Topology" (has statements but very few proofs)
• Notes of Haiman on derived categories and functors (this material in not included in Chao Li's notes, but we did do it in class)
• Notes of de Cataldo from recent lectures in Ann Arbor.
• BBD
• Notes of Williamson on KL theorems

Prerequisites: For the first half of the class, it will be necessary to be comfortable with notions from algebraic topology (e.g., cohomology, Poincare duality, the first Chern class) and have some familiarity with basic notions from complex algebraic geometry. It would probably be good to have seen sheaves in one context or another before this class.

For the applications to representation theory towards the latter half of the course, the prerequisites include a familiarity with complex representations of finite groups and finite dimensional representations of semi-simple complex Lie algebras.

Grading: Undergraduates or graduate students wishing to take this course for a grade should speak with the instructor within the first week of classes.

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