Seminar: Perverse sheaves (2022)

Goal:

The goal of this seminar is to learn about perverse sheaves and the geometric Satake equivalence. Instead of seminar talks, we will hold weekly meetings/discussions over assigned readings and complete exercises. We plan on keeping a shared, online journal amongst ourselves to highlight lingering questions and gather insights and thoughts that struck us during the readings. We also plan on keeping an online working document of partially/fully tex-ed up solutions.

Materials:

• P. Achar, "Perverse sheaves and its applications in representation theory"

• M. Goresky, lecture notes on perverse sheaves (here)

• Supplementary materials and helpful notes will be gathered along the way and published onto this website.

Space-time coordinates:

• Time: Fridays, 12-2pm

• Venue: Math lounge 

Schedule:

• 11/18: Chapter 1, sheaf theory (Marielle)
  Summary: Ran through this chapter briefly, summarized the different functors and base-change theorems.

• 11/25: Chapter 1, sheaf theory (Together)
  Summary: Completed some exercises from Chapter 1

• 12/2: Chapter 1, sheaf theory (Marielle)
  Summary: Computations of push-forward of sheaves in Section 1.10
  Break

• 01/13: Chapter 2, constructible sheaves (Emma)
  Summary: orientation sheaf, basic definitions of stratification and constructibility.

• 01/20: Chapter 2.8, Verdier duality and proof (Marielle)

• 01/27: Chapter 2.11, Homology and Fundamental classes (Emma)
  Summary: Boore-Moore homology

• 02/04: Chapter 3.1, Perverse sheaves (Marielle)
  Summary: Appendix A.7 (generalities of t-structures), proving that the perverse t-structure is a t-structure

• 02/11: Computing examples of perverse sheaves (Yingdi)
  Summary: For the inclusion j of a point into C, we computed j_! and j_* of the constant sheaf