Algebraic geometry, computational algebra
My main areas of interest are computational algebra and algebraic geometry, commutative algebra, and algebraic geometry. My original interest in computational methods was their application to problems in algebraic geometry. Since then, my work has proceeded in several related directions. I have studied the complexity of the algorithms (mainly Gröbner bases). 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). In the last few years, Peeva and I have been interested in Hilbert schemes: classical ones, toric Hilbert schemes, and parameter spaces over the exterior algebra.
- 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.