Vanishing cycles and perverse sheaves
MATH-658 / 1 credit
Teacher(s): Hemelsoet Nicolas Thierry Nathalie, Testerman Donna
Language: English
Frequency
Every year
Summary
This course will explain the theory of vanishing cycles and perverse sheaves. We will see how the Hard Lefschetz theorem can be proved using perverse sheaves. If we have more time we will try to see the decomposition theorem and how to categorify certain perverse sheaves.
Content
The purpose of this course is both to understand original motivation by Lefschetz to study vanishing cycles, and some more recent development (work by Kapranov and Schechtman to categorify perverse sheaves, but other topics are possible). We will mostly focus on examples rather than the theory.
The chapters will be as follow :
Chapter 1 : Topology of complex algebraic varieties and vanishing cycles.
Chapter 2 : The six functor formalism for constructible sheaves.
Chapter 3 : Perverse sheaves, definition and examples (perverse sheaves on the disk).
Chapter 4 (if time) : the decomposition theorem.
Chapter 5 (if time) : Sketch of the proof of Hard Lefschetz.
Chapter 6 (if time) : Perverse schobers.
During exercices, we will mostly try to understand examples and do concrete calculations.
Keywords
perverse sheaves, algebraic geometry, topology, derived category, representation theory
Learning Outcomes
By the end of the course, the student must be able to:
- Integrate the formalism of perverse sheaves, the statement of the main results and examples.
Resources
Bibliography
The main reference we will follow is : "An illustrated guide to perverse sheaves" by G. Williamson (for chapter 2 and 3).
Références suggérées par la bibliothèque
Moodle Link
In the programs
- Exam form: Oral presentation (session free)
- Subject examined: Vanishing cycles and perverse sheaves
- Lecture: 11 Hour(s)
- Exercises: 6 Hour(s)
- Type: optional