Multigraded commutative algebra and the geometry of syzygies

代数几何是研究作为多项式方程组解集出现的空间；这样的空间在数学和科学中起着重要的作用。代数几何中的一个基本问题是：这样一个空间的几何告诉我们关于决定它的多项式的什么？PI研究的首要目标是使用计算代数中的技术来研究这一主题的开放问题。这项研究将导致开源计算代数系统Macaulay2的新软件的开发。PI还将与奥本大学的退伍军人资源中心合作，继续向奥本大学的数学退伍军人伸出援手。PI将采用从射影几何到环面几何的协同几何技术。特别是，PI将使用交换代数中的技术，在Berkesch-Erman-Smith和Orlov关于环变同调性质的猜想上取得进展。PI也将推广，从投影到加权投影设置，一个著名的格林定理关于曲线嵌入投影空间的自由分辨率的线性。在第三个项目中，PI将通过推广适用于射影空间上的束的Eisenbud-Fløystad-Schreyer算法，开发一种有效的算法来计算光滑射影环上的束上同调。PI还将解释在完整交叉点上自由分辨率的拟合理想的周期性现象，利用艾森勃-皮耶娃在这种分辨率结构上的工作。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.