I study symplectic geometry, the natural geometry of classical and quantum mechanics, where coordinates come in pairs corresponding to position and momentum. Replacing "phase spaces" with more abstract "symplectic manifolds," the laws of physics translate into a way to measure 2-dimensional areas. I study problems at the interface of symplectic geometry, algebraic geometry, and discrete geometry. Most recently, I have been trying to identify and quantify the essential features of a symplectic structure and of the symmetries that preserve such a structure. With my collaborators, I have uncovered intriguing patterns in the quantitative data arising in our investigations. We have been able to verify a number of these patterns and there remain many open questions.
- The topology of toric origami manifolds (with Ana Rita Pires), Math. Research Letters, 20 no. 5 (2013), 885–906.
- Orbifold cohomology of torus quotients (with Rebecca Goldin and Allen Knutson), Duke Math. J. 139 no. 1 (2007), 89–139.
- Computation of generalized equivariant cohomologies of Kac-Moody flag varieties (with Megumi Harada and Andre Henriques), Adv. in Math. 197 no. 1 (2005), 198–221.
- Conjugation spaces (with Jean-Claude Hausmann and Volker Puppe), Algebr. Geom. Topol. 5 (2005), 923–964.
- Distinguishing chambers of the moment polytope (with Rebecca Goldin and Lisa Jeffrey), J. Symp. Geom. 2 no. 1 (2003), 109–131.