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 combinatorics (of polytopes, hyperplane arrangements and matroids and...) and representation theory (of algebraic groups, reflection groups, Lie algebras and...).
Much of my work involves finding explicit 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. At the most basic level one can ask for the dimensions of local or global intersection cohomology. Finer information is obtained by computing the intersection cohomology canonically as vector spaces, while in some cases it is possible to understand the structure of the entire category of perverse sheaves (for a fixed stratification).
In some of these situations the combinatorial data can make sense even when there is no algebraic variety. For example, non-rational polytopes and fans do not correspond to toric varieties, and non-crystallographic reflection groups do not correspond to flag varieties. In those cases the combinatorial computation of intersection cohomology can still be carried out, although new proofs are needed to show that they work as expected.
Perverse sheaves on symmetric products of the plane, with Carl Mautner.
Singular Hodge theory for combinatorial geometries, with June Huh, Jacob Matherne, Nicholas Proudfoot and Botong Wang.
What is ... the Dowling-Wilson conjecture?, with Jacob Matherne and Nicholas Proudfoot, Notices Amer. Math. Soc. 79 (2024), no. 9, 1197-1201.
A semi-small decomposition of the Chow ring of a matroid, with June Huh, Jacob Matherne, Nicholas Proudfoot and Botong Wang, Adv. Math. 409 (2022), Paper No. 108646, 49 pp.
Kazhdan-Lusztig polynomials of matroids under deletion, with Artem Vysogorets, Electron. J. Combin. 27 (2020), no. 1, P1.17.
Ringel duality for perverse sheaves on hypertoric varieties, with Carl Mautner, Adv. Math. 344 (2019), 35-98.
Matroidal Schur Algebras with Carl Mautner, J. Alg. Comb. 46 (2017), no. 1, 51-75.
Quantizations of conical symplectic resolutions II: category O and symplectic duality with Anthony Licata, Nicholas Proudfoot and Ben Webster, and an appendix by Ivan Loseu, Astèrisque 384 (2016), 75-179.
Quantizations of conical symplectic resolutions I: local and global structure, with Nicholas Proudfoot and Ben Webster, Astèrisque 384 (2016), 1-73.
Hypertoric category O, with Anthony Licata, Nicholas Proudfoot and Ben Webster. Adv. Math. 231 (2012), no. 3-4, 1487-1545.
Appendix to Modular intersection cohomology complexes on flag varieties, by Geordie Williamson, Math. Zeitschrift 272 (2012), nos. 3-4, 697-727.
Localization algebras and deformations of Koszul algebras, with Anthony Licata, Christopher Phan, Nicholas Proudfoot and Ben Webster. Selecta Math. 17 (2011), no. 3, 533-572.
Gale duality and Koszul duality, with Anthony Licata, Nicholas Proudfoot, and Ben Webster. Adv. Math. 225 (2010), no. 4, 2002-2049.
The hypertoric intersection cohomomology ring, with Nicholas Proudfoot. Invent. Math. 177 (2009), no. 2, 337-379.
The equivariant Chow rings of quot schemes, with Linda Chen and Frank Sottile. Pacific J. Math. 238 (2008), no. 2, 201-232.
Koszul duality for toric varieties. Trans. Amer. Math. Soc. 359 (2007), 385-415.
Remarks on the combinatorial intersection cohomology of fans, Pure and Applied Mathematics Quarterly 2 (2006), no. 4, 1149-1186.
Equivariant-constructible Koszul duality for dual toric varieties, with Valery Lunts. Adv. Math. 201 (2006), no. 2, 408-453
Lower bounds for Kazhdan-Lusztig polynomials from patterns, with Sara Billey. Transform. Groups 8 (2003), no. 4, 321-332.
Hyperbolic localization of intersection cohomology. Transform. Groups 8 (2003), no. 3, 209--216.
Perverse sheaves on Grassmannians. Canad. J. Math. 54 (2002), no. 3, 493-532.
On the reducibility of characteristic varieties. Proc. Amer. Math. Soc. 130 (2002), no. 7, 2037-2043.
From moment graphs to intersection cohomology, with Robert MacPherson. Math. Ann. 321 (2001), no. 3, 533-551.
Intersection homology of toric varieties and a conjecture of Kalai, with Robert MacPherson. Comment. Math. Helv. 74 (1999), no. 3, 442-455.
Perverse sheaves on rank stratifications, with Mikhail Grinberg. Duke Math. J. 96 (1999), no. 2, 317-362.