For specifics, here is my research statement.
Published and submitted papers:
- Rigid monomial ideals, joint with Sonja Mapes, submitted.
- A minimal poset resolution of stable ideals, to appear in Progress in Commutative Algebra: Ring Theory, Homology, and Decompositions,, Ed. by Sean Sather-Wagstaff, Christopher Francisco, Lee Klingler and Janet C. Vassilev. Here is a preprint version of the paper.
- Poset resolutions and lattice-linear monomial ideals, Journal of Algebra, Volume 323, Issue 4, 15 February 2010, Pages 899-919.
Work in progress:
- CW complexes and poset resolutions, joint with Alexandre Tchernev, in progress.
- Uniform matroids and multigraded resolutions, joint with Amanda Beecher and Alexandre Tchernev, in progress.
- Poset homology and rigid monomial ideals, joint with Sonja Mapes, in progress. Here's a talk I gave about the project.
Problems I'm interested in:
- Develop an encyclopedia between properties of finite simple graphs and properties of finite atomic lattices via edge ideals.
- Give a combinatorial characterization of posets which support exact complexes of vector spaces.
- Classify minimal resolutions of monomial ideals by automorphism type.