Research - the questions that guide us

We work at the crossroad of geometry, combinatorics, and algebra. For example, we relate configurations of geometric shapes (such as lines, points, and circles) with algebraic objects (such as polynomials) and combinatoric objects (such as polygons and directed graphs). Specific projects include the following:

Families of algebraic varieties. Algebraic geometry studies the geometry of the zero sets of polynomials. Varying a polynomial's coefficients yields an algebro-geometric object. Therefore, our guiding questions are: What possible zero sets (or isomorphism classes) occur by moving such coefficients towards a limit? What is the geometry of the space parametrizing all zero sets and their limits? We also use such projects to introduce students to moduli theory via the development of combinatorial and computational tools.

Polytopes, directed graphs, and families of linear maps. Given an acyclic directed graph, if one considers all nonnegative flows on the edges subject to a boundary condition at the vertices, one arrives at a polytope parametrizing all such flows. Such a setting determines a classification problem because there are only finite polytopes for a given graph. We are interested in families of polytopes of dimensions larger than three and their connections to algebraic geometry.

Publications

  1. Variation of stability for moduli spaces of unordered points in the plane (with B. Schmidt) Submitted.

  2. Algebraic and analytic compactifications of moduli spaces (with M. Kerr). To appear the Notices of the AMS.

  3. Quivers and moduli of their thin sincere representations in Macaulay2 (with M. Barker) Submitted.

  4. The Fulton-MacPherson compactification is not a Mori dream space (with José Luis González, Evangelos Routis) Submitted.

  5. Geometric interpretation of toroidal compactifications of moduli of points in the line and cubic surfaces (with M. Kerr and L. Schaffler) Advances in Mathematics, 381, 2021.

  6. Applications of the moduli continuity method to log K-stable pairs (with J. Martinez-Garcia and C. Spotti)Journal of the London Mathematical Society 103.2 (2021): 729-759.

  7. Moduli of cubic surfaces and their anticanonical divisors (with J. Martinez-Garcia) Revista Matematica Complutense (RMC) 32.3 (2019): 853-873

  8. Modular interpretation of a non-reductive Chow quotient (with N. Giansiracusa). Proceedings of the Edinburgh Mathematical Society.

  9. Wonderful compactifications of the moduli space of points (joint with E. Routis). European Journal of Mathematics, 3(3), 520-564.

  10. VGIT for pairs, a computational approach (with J. Martinez-Garcia). Proceeding of the AMS.

  11. Families of elliptic curves in Projective space and Bridgeland Stability (with C. Lozano-Huerta, and B. Schmidt) Michigan Mathematical Journal.

  12. On the GIT quotient of quintic surfaces. Transactions of the American Mathematical Society. 371 (2019), 4251-4275.

  13. A slice of the moduli space of lines arrangements (joint with K. Ascher) Algebra and Number theory.

  14. Compactifications of the moduli space of plane quartics and two lines (with Z. Zhang and J. Martinez-Garcia) European Journal of Mathematics, 2018, Volume 4, Issue 3.

  15. On the neighborliness of thin-sincere representations of quivers (with D. Mckenzie)

Software and computational tools

1. Variations of GIT quotients package. In joint work with Jesus Martinez-Garcia (University of Essex, UK), we developed software, in Python, that calculates the computational information required to describe the GIT quotients parametrizing a pair defined by hypersurface and a hyperplane.

2. Quivers and their thin sincere representations. In work with Mary Barker (WashU), we developed a Macaulay2 package to describe their geometry and moduli spaces of thin-sincere representations of acyclic quivers i.e., the dimension vector is all equal to one.