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. Much of my work involves finding explicit combinatorial methods to compute intersection cohomology and perverse sheaves on these spaces, in terms of combinatorial data provided by the symmetries and orbits of a group action.
MG: computing with moment graphs
Some time ago I wrote some Macaulay 2 code to compute (torus equivariant) intersection cohomology (and ordinary cohomology) for certain nice spaces including Schubert varieties, in terms of the "moment graph" or "1-skeleton" formed by the zero and one-dimensional orbits. The code is available here. It's far from a finished system, and sometimes I have had to make modifications or extensions for a particular computation. If there's something you would like to do with it that it doesn't currently do, let me know.
- Slides from a talk at a meeting "Sheaves in representation theory" on the Isle of Skye, Scotland, in May 2010.
- An extended abstract from an Oberwolfach conference in toric geometry, January 2009.
- Slides from a talk at NYU AMS meeting, Spring 2008.
- Slides from a talk at Rutgers AMS meeting, Fall 2007.
Seminars and conferences
Tara Holm and I
organized a special session "Combinatorial
techniques in equivariant topology" at the Fall 2006 AMS
meeting in Storrs, CT. Follow the link for abstracts and slides for
many of the talks.