I’ve been too lazy to write in detail about the progress in my research (well, I am writing six papers and applying to jobs, so it isn’t entirely due to laziness), but I did recently speak in the symplectic seminar at MIT, and have posted the slides on my webpage. Obviously, they’re less useful without someone to explain them, but given the current lack of an overarching paper on the subject (that’s no. 5 on the list, I promise), I thought it might be edifying. Executive summary below the cut.Tom Braden, Tony Licata, Nick Proudfoot and I have been working on understanding a certain kind of duality between holomorphic symplectic manifolds. At the moment, this lacks any really good overarching mathematical definition, but we have some really cool examples:
- the cotangent bundle to the flag variety is dual to the cotangent bundle to the Langlands dual flag variety .
- the Hilbert scheme of points on is self-dual.
- the quiver variety of two dominant weights of a finite-dimensional, simply-laced Lie algebra is dual to the slice in to the orbit in the affine Grassmannian of said Lie algebra (it’s probably better to say Langlands dual, but that’s the same, since my algebra is simply-laced).
One point I got to at the end, but couldn’t cover in nearly the detail I would have liked (in part because I haven’t worked out the details) is the implications of this for knot theory. In particular, this seems to explain why Stroppel on one hand, and Seidel-Smith on the other, got completely different geometric definitions of the same knot homology theory; they were working in geometric situations (cotangent bundles of Grassmannians on one hand, and Slodowy slices in nilcones on the other) which are dual under this prescription.
All right, I’d better get back to writing the actual paper on this stuff, so that people have something more detailed to read than my slides.