Studies on Local Cohomology, Derivations, Integral Dependence, and Blowup Algebras

这个项目是在交换代数中进行的，但也受到了代数几何的启发。多元多项式方程组在科学和技术中有着广泛的应用，例如在工程、计算机科学、密码学、编码理论、机器人学、模式识别和理论物理中。交换代数和代数几何致力于这种多项式方程组的定性研究。这个项目的重点是代数方法。该项目的目标之一是了解与多项式方程组的解集相切的矢量。另一个目标是找到描述给定几何对象的多项式方程组。后者在计算机图形学、代数统计和结构刚度方面有应用。PI打算让本科生、研究生和博士后研究员参与她的研究。她将继续组织国家和国际会议。在她的整个活动中，国际数学联合会将继续促进数学中代表性不足的群体。本研究的第一个目的是将射影空间上向量场的度与它们保持不变的曲线或簇的不变量联系起来。这是一个已经研究了一个多世纪的难题，主要从复分析、动力系统、代数和微分几何的角度进行研究。PI将使用交换代数中的工具来研究这个问题。第二个目标是找出射影空间中的一个簇是集合论完全交的准则。当只有一个非零局部上同调模时，PI相信集合论完全交的性质编码在这个模的结构中。第三个目标是利用交集理论中出现的重数概念来证明模的积分依赖性的一个数值刻画。这类定理与等奇性理论有关，事实上，它们导致了一族解析空间的纤维数值条件是惠特尼等奇性的，因此在拓扑上是平凡的。最后一个目标是研究定义射影空间之间有理映射的图形和图像的隐式方程。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project is in commutative algebra, but has also been inspired by algebraic geometry. Systems of polynomial equations in several variables occur in many applications in science and technology, for instance in engineering, computer science, cryptography, coding theory, robotics, pattern recognition, and theoretical physics. Commutative algebra and algebraic geometry are concerned with the qualitative study of such systems of polynomial equations. This project focuses on the algebraic approach. One of the goals of the project is to understand the vectors that are tangent to solution sets of systems of polynomial equations. Another objective is to find systems of polynomial equations describing a given geometric object. The latter has applications to computer graphics, algebraic statistics, and rigidity of structures. The PI intends to involve undergraduate students, graduate students, and postdoctoral fellows in her research. She will continue to organize national and international meetings. Throughout her activities, the PI will continue to promote underrepresented groups in mathematics. The first objective of this research is to relate degrees of vector fields on projective space to invariants of curves, or varieties, that they leave invariant. This is a difficult problem that has been studied for over a century, mainly from the point of view of complex analysis, dynamical systems, algebraic and differential geometry. The PI will investigate this question using tools from commutative algebra. The second objective is to find criteria for a variety in projective space to be a set-theoretic complete intersection. When there is only one non vanishing local cohomology module the PI believes that the property of being a set-theoretic complete intersection is encoded in the structure of this module. The third objective is to prove a numerical characterization of integral dependence of modules using a notion of multiplicity that arises in intersection theory. Theorems of this kind have a bearing on equisingularity theory, in fact they lead to fiber-wise numerical conditions for a family of analytic spaces to be Whitney equisingular, hence topologically trivial. The last objective is to study the implicit equations defining the graph and the image of rational maps between projective spaces.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.