Multigraded commutative algebra and the geometry of syzygies

Algebraic geometry is the study of spaces that arise as solution sets to systems of polynomial equations; such spaces play an important role throughout mathematics and the sciences. A fundamental question in algebraic geometry is: what does the geometry of such a space tell one about the polynomials that determine it? The overarching goal of the PI’s research is to use techniques in computational algebra to study open problems on this theme. This research will lead to the development of new software for the open-source computational algebra system Macaulay2. The PI will also continue his outreach to veterans in mathematics at Auburn University, in collaboration with the university’s Veterans Resource Center. The PI will adapt the techniques of the geometry of syzygies from projective geometry to toric geometry. In particular, the PI will use techniques in commutative algebra to make progress on conjectures of Berkesch-Erman-Smith and Orlov on the homological properties of toric varieties. The PI will also generalize, from the projective to the weighted projective setting, a celebrated theorem of Green on the linearity of free resolutions of curves embedded in projective space. In a third project, the PI will develop an efficient algorithm for computing sheaf cohomology over smooth projective toric varieties by generalizing an algorithm due to Eisenbud-Fløystad-Schreyer that applies to sheaves on projective space. The PI will also explain a periodicity phenomenon for the Fitting ideals of free resolutions over complete intersections, leveraging work of Eisenbud-Peeva on the structure of such resolutions. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.