video of the talk

Around 2007, Braden, Licata, Proudfoot and I came up with the slightly daft idea that certain singular symplectic varieties come in dual pairs whose definition we could not explain, with relations between them that we struggled to articulate, let alone prove. By 2014, we were able to produce at least a coherent list of properties we expected to match in dual pairs, the most striking being a Koszul duality between associated category O's.

This was either excellent or terrible timing (depending on your perspective), since immediately afterwards Braverman-Finkelberg-Nakajima produced a uniform construction of a huge number of examples, including almost all of those known to us. I'll take this talk as an opportunity to organize where we stand on the relationship between our work and BFN's, as well as the copious follow-up work which has appeared since their paper.