GAAG seminar - Cohomology of symmetric stacks

I will talk about a joint project with Chenjing Bu, Ben Davison, Andrés Ibáñez-Núñez, and Tasuki Kinko.
For a large class of stacks, we decompose their cohomology in BPS cohomology, which is a structure originating in enumerative geometry of Calabi-Yau 3-folds, but which is of interest beyond this class of examples. 
Such stacks include smooth stacks (such as the moduli of G-bundles on a curve), symplectic stacks (such as the moduli of G-Higgs bundle on a curve), or (-1)-shifted symplectic stacks (such as the moduli of semistable sheaves on a Calabi-Yau threefold). 
In the pre-talk, I plan to discuss the local structure of the moduli of semistable vector bundles on a curve, and properties of vanishing cycles.  
In the talk, I will explain the proof of the main theorem for the example of rank 2 and degree 0 semistable vector bundles on a curve (which was already treated by Meinhardt and Mozgovoy-Reineke). I will then introduce BPS cohomology and state the theorem for (-1)-shifted and 0-shifted symplectic stacks. Time permitting, I will explain conjectural applications of BPS cohomology in Langlands duality for compact real oriented 3-manifolds (following Ben-Zvi-Gunningham-Jordan-Safronov). BPS cohomology can be also used to state a version of topological mirror symmetry for G-Higgs bundles (for general reductive groups G).