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

这个项目是交换代数，但也受到代数几何的启发。多个变量的多项式方程系统在科学和技术的许多应用中出现，例如工程、计算机科学、密码学、编码理论、机器人、模式识别和理论物理。交换代数和代数几何关注的是这种多项式方程组的定性研究。这个项目着重于代数方法。该项目的目标之一是理解与多项式方程组解集相切的向量。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。该项目计划邀请本科生、研究生和博士后参与她的研究。她将继续组织国内和国际会议。在她的活动中，PI将继续促进数学中代表性不足的群体。本研究的第一个目标是将射影空间上向量场的度数与曲线的不变量或变化联系起来，使它们保持不变。这是一个已经被研究了一个多世纪的难题，主要是从复杂分析、动力系统、代数和微分几何的角度来研究的。PI将使用交换代数中的工具来研究这个问题。第二个目标是在射影空间中找到一个变体是集合论完全交集的准则。当只有一个不消失的局部上同模时，PI认为该模的结构中包含了集论完全交的性质。第三个目标是用在交点理论中出现的多重性的概念证明模的积分依赖性的数值表征。这类定理与等奇异性理论有关，事实上，它们导致了一组解析空间是惠特尼等奇异的纤维型数值条件，因此拓扑平凡。最后一个目的是研究定义投影空间间有理映射的图和象的隐式方程。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。

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.