I study the topology of algebraic varieties, especially their singularities.  Most of my work involves intersection cohomology and more general perverse sheaves.  For many interesting spaces, such as flag varieties and toric varieties and their generalizations, intersection cohomology and perverse sheaves give important information about questions in representation theory and combinatorics.

My current primary research project, joint with Anthony Licata, Nicholas Proudfoot, and Ben Webster, concerns "symplectic duality", which relates certain pairs of singular affine varieties endowed with a symplectic resolution and a torus action.  Examples of such pairs include the nilpotent cone of SL(n) (which is self-dual), pairs of affine hypertoric varieties determined by Gale dual hyperplane arrangements, and Nakajima quiver varieties for finite Dynkin quivers, which are dual to slices to Schubert varieties in an affine Grassmannian.  The duality between such pairs should manifest in a number of ways, including duality on a deformation of the cohomology of the resolutions which respects filtrations coming from the decomposition theorem, and a Koszul duality relating certain associated abelian categories which generalize the Bernstein-Gelfand-Gelfand category O from representation theory.

My CV (pdf format)            My publications               

MG:  Macaulay 2 code to compute equivariant (intersection) cohomology with moment graphs
 
Working seminar on quiver varieties and representation theory, Spring 2006

Slides from my talk at the Toric Topology conference, Osaka, June 2006.
Slides from a talk at Rutgers AMS meeting, Fall 2007.
Slides from a talk at NYU AMS meeting, Spring 2008.
An extended abstract from an Oberwolfach conference in toric geometry, January 2009.