Collaborative Research: Differential Methods, Implicitization, and Multiplicities with a View Towards Equisingularity Theory

这项研究项目涉及几何对象的结构，这些几何对象是作为多变量多项式方程组的解集出现的。这种被称为变种的对象在整个数学以及在科学和工程中的应用中都发挥着基础性的作用。受数学家Henri Poincaré在1891年提出的一个问题的启发，研究人员将研究变种的局部性质和变种点处切线的全局特征之间的关系，因为这些点不同。他们还将致力于隐含化，这是纯数学中的一个经典问题，对几何建模和计算机辅助设计的科学家非常感兴趣。给定任何几何对象，例如曲线或曲面，目标是找到将该几何对象作为解集的多项式方程组；了解这些“隐式”方程有助于深入了解该几何对象。PIS将让研究生和博士后访客参与到这个项目中来，他们将通过组织国际项目、会议和国家在线研讨会来促进科学交流。研究人员将致力于与代数向量场、Rees环的隐含化问题、等奇点理论和剩余交集有关的项目。PI将使用交换代数中的工具来研究奇点的类型和变种的全局不变量如何反映在与变种相切的矢量场的属性中。特别地，他们希望建立射影平面曲线奇点的类型和星座与其齐次坐标环的导子模的分次Betti数之间的对应关系。确定定义射影空间间有理映射的图形和映象的隐式方程是消去论中的经典问题。PI将集中在Cremona映射的情况下，其中图形的隐式方程还提供逆映射的参数化。对于显性有理映射，他们将研究有理映射的投影度与定义图的方程的个数和二次数之间的关系。在等奇性理论中，人们寻求基于纤维多重性的准则，以使一族分析空间是惠特尼等奇性的，因此在拓扑上是平凡的。PI计划通过使用一个受交集理论启发的多重性的新概念，为具有任意奇点的解析空间设计这样一个判据。研究生将作为项目的一部分得到支持。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 research project concerns the structure of geometric objects that arise as solution sets of systems of polynomial equations in several variables. Such objects, called varieties, play a fundamental role throughout mathematics as well as in applications in science and engineering. Motivated by a question that originates with mathematician Henri Poincaré in 1891, the investigators will study the relationship between local properties of a variety and global features of tangents at points of the variety, as these points vary. They will also work on implicitization, a classical question in pure mathematics that is of much interest to scientists in geometric modelling and computer aided design. Given any geometric object, such as a curve or surface, the goal is to find the system of polynomial equations that has the geometric object as a solution set; knowing these 'implicit' equations provides insight into the geometric object. The PIs will involve graduate students and postdoctoral visitors in the project, and they will facilitate scientific exchange by organizing international programs, conferences, and national online seminars.The investigators will work on projects pertaining to algebraic vector fields, the implicitization problem for Rees rings, equisingularity theory, and residual intersections. The PIs will use tools from commutative algebra to investigate how the types of singularities and the global invariants of a variety are reflected in properties of the vector fields that are tangent to the variety. In particular, they wish to establish a correspondence between the types and constellation of the singularities of a projective plane curve on the one hand and the graded Betti numbers of the module of derivations of its homogeneous coordinate ring on the other. Determining the implicit equations defining the graph and image of a rational map between projective spaces is a classical problem in elimination theory. The PIs will concentrate on the case of Cremona maps, where the implicit equations of the graph also provide a parametrization of the inverse map. For dominant rational maps they will investigate the relationship between the projective degrees of the rational map and the number and bidegrees of the equations defining the graph. In equisingularity theory, one seeks fiberwise multiplicity-based criteria for a family of analytic spaces to be Whitney equisingular and hence topologically trivial. The PIs plan to devise such a criterion for analytic spaces with arbitrary singularities by using a new notion of multiplicity inspired by intersection theory. Graduate students will be supported as part of the project.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.