timothy clark

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My current research is focused on explicitly writing down exact complexes of multigraded modules (and free resolutions of multigraded modules) whose mapping structure mirrors an appropriately chosen partially ordered set. My favorite method of making this connection is to create an exact complex of vector spaces using the combinatorics, topology and homology of the given poset. Finding a partially ordered set that submits to this process and relates to a given multigraded module is not a trivial task!

For specifics, here is my research statement.

Published and submitted papers:

3. Rigid monomial ideals, joint with Sonja Mapes, submitted.
2. 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.
1. 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.