Research Focus
Computational algebraic geometry, computational algebra, applications to physics, biology, and other fields.
My main areas of interest are computational algebraic geometry, algebraic geometry, commutative algebra, and applications of computational methods to algebraic geometry and other fields. All of my research is in some way related to computation in algebraic geometry and related fields.
I am a principal author of the Macaulay2 computer algebra system (written with Dan Grayson, now many others are involved too). I have been developing algorithms for computing in commutative algebra and algebraic geometry (e.g. computing with line bundles, computing Hilbert functions, free resolutions, sheaf cohomology, computing with Hilbert schemes and Quot schemes, computing with Calabi-Yau varieties, and computing with toric varieties).
Over the last several years, I have been collaborating with Liam McAllister (Cornell Physics) and others in string theory applying computational methods in algebraic geometry, commutative algebra, and toric geometry to problems in theoretical physics, especially string theory.
I am interested in applications of computer algebra methods to problems in biology, for instance the recent paper with Heather Harrington (MPI Dresden) and Alan Veliz-Cuba (University of Dayton).
I also am interested in the application of (computational) algebraic geometry methods to study Kuramoto oscillators, with two papers, one with Steve Strogatz and Alex Townsend (both at Cornell), and another with Heather Harrington and Hal Schenck (Auburn Univ).
On the pure algebraic geometry side, I have been working with Roy Skjelnes (KTH) and Greg Smith (Queen's University) on understanding the structure of Quot schemes on projective spaces. In certain cases we can determine exactly which ones are smooth varieties.
Publications
- Algebraic geometry of Bayesian networks (with L. Garcia and B. Sturmfels), preprint (2003).
- Toric Hilbert schemes (with I. Peeva), Duke Math. J. 111 (2002), 419–449.
- Computations in Algebraic Geometry with Macaulay 2 (D. Eisenbud, D. Grayson, M. Stillman, B. Sturmfels, eds.), Springer, 2001.
- Computing sheaf cohomology on toric varieties (with D. Eisenbud and M. Mustata), J. Symbolic Computation 29 (2000), 583–600.
- A criterion for detecting m-regularity (with D. Bayer), Invent. Math. 87 (1987), 1–11.
