Cohomology of symmetric stacks
Frequency
Only this year
Summary
This is a reading seminar on the preprint "Cohomology of Symmetric stacks" by Chenjing Bu, Ben Davison, Andrés Ibáñez Núñez, Tasuki Kinjo and Tudor Padurariu. Every session consists of a 1h30 talk by participants and invited speakers (Sarunas Kaubrys, Tanguy Vernet).
Content
The reading seminar on 'cohomology of symmetric stacks' follows last year's reading group on Stacks and 'Intersection Cohomology and Perverse Sheaves', however it will be self-contained as we will recall the relevant background.
The goal is to understand how to obtain decompositions of
- the intersection cohomology of good moduli spaces of smooth stacks, such as moduli spaces of G-bundles on a curve
- the BPS cohomology of g.m.s. of (-1)-shifted symplectic stacks such as loop stacks of G-Higgs bundles on a curve.
The plan is as follows :
Week 1 - Introduction
Week 2 - Component lattices
Week 3 - Reviews of Stacks and symmetric stacks
Week 4 - Shifted symplectic structures (external speaker : Sarunas Kaubrys)
Week 5 - Reviews of perverse sheaves and monodromic mixed Hodge modules, equivariant cohomology
Week 6 - Orientations
Week 7 - BPS sheaves
Week 8 - Cohomological Hall Induction
Week 9 - Proof of cohomological integrality theorems (external speaker : Tanguy Vernet)
Week 10 and 11 - Flash talks by all participants on various applications (quotient stacks, linear stacks, G-Higgs bundles...)
Keywords
Stacks, Moduli spaces, BPS sheaf, integrality theorems
Learning Prerequisites
Required courses
Algebraic Geometry III, Homological algebra. Some knowledge on stacks will help.
Learning Outcomes
By the end of the course, the student must be able to:
- Identify the construction of component lattices, BPS sheaf and the statements of integrality theorems for smooth or (-1)-shifted symplectic stacks and its relevance for applications such as Langlands duality for 3-manifolds.
Dans les plans d'études
- Forme de l'examen: Exposé (session libre)
- Matière examinée: Cohomology of symmetric stacks
- Cours: 16 Heure(s)
- Exercices: 16 Heure(s)
- Type: optionnel