Ben Webster: Research
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The photo: Jupiter, a bar in Berkeley.
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My research is on various aspects of the interaction between
representation theory, algebraic geometry and topology. Things I've been
thinking about recently include:
- My primary research topic at the moment is understanding
connections between the geometry of symplectic varieties and the
representation theory of their deformation quantizations (roughly
modeled on branch of representation theory which understands
representations as D-modules). This includes
- Exploring generalizations of the Beilinson-Bernstein localization theorem.
- Using this as a tool for exploring a conjectural duality between certain holomorphic symplectic varieties. (joint with Nick Proudfoot, Tony Licata and Tom Braden)
- Investigating connections between geometry of Springer fibers/Slodowy slices and knot homology/representation theory of category O/finite W algebras. (joint with Catharina Stroppel).
- Carrying out this program for hypertoric varieties, with an emphasis on understanding relations between Gale dual hypertoric varieties. (joint with Nick Proudfoot, Tony Licata and Tom Braden)
- Geometric knot homology:
- This includes studying how triply graded knot homologies can be described geometrically, by computing the Hochschild homology of Soergel bimodules from a geometric perspective. I'm working on giving geometric proofs of invariance and decategorification for colored HOMFLY homology, where no algebraic proofs currently exist. (joint with Geordie Williamson)
- I'm also interested in using certain categories of representations of deformation quantizations of quiver varieties to categorify Reshetikhin-Turaev invariants.