Abstract:In recent work with Braden, Licata and Proudfoot, we showed that certain algebras constructed from hyperplane arrangements have a number of nice properties which are surprisingly reminiscent of the BGG category O; in particular, they are Koszul, and Koszul duality corresponds to a well known combinatorial duality. I'll explain why we think properties are connected to a geometric origin for both these categories, and how this suggests an underlying duality between pairs of symplectic varieties.
In a second talk, I will go into more detail about cohomological and categorical relationships (some proven, some conjectural) between dual varieties, the use of these varieties in categorification and their proposed applications in knot theory and representation theory.