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

本研究项目涉及几何对象的结构，这些几何对象是由多个变量的多项式方程系统的解集组成的。这些对象被称为变种，在整个数学以及在科学和工程中的应用中起着重要作用。受1891年数学家亨利·庞加莱（Henri poincar<s:1>）提出的一个问题的启发，研究人员将研究多样性的局部性质与多样性点上切线的整体特征之间的关系，因为这些点在变化。他们还将研究隐式化，这是纯数学中的一个经典问题，几何建模和计算机辅助设计领域的科学家对此非常感兴趣。给定任何几何对象，如曲线或曲面，目标是找到多项式方程系统，该系统将几何对象作为解集；了解这些“隐式”方程有助于深入了解几何对象。pi将包括研究生和博士后访问学者，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。研究人员将研究与代数向量场、里斯环的隐化问题、等奇异性理论和剩余交集有关的项目。pi将使用交换代数的工具来研究一个变量的奇异类型和全局不变量是如何反映在与该变量相切的向量场的性质上的。特别地，他们希望在一个射影平面曲线的奇点的类型和星座与其齐次坐标环的导数模的梯度Betti数之间建立对应关系。确定定义投影空间间有理映射的图象的隐式方程是消去理论中的一个经典问题。pi将集中在Cremona映射的情况下，其中图的隐式方程也提供了逆映射的参数化。对于主导有理图，他们将研究有理图的投影度数与定义图的方程的数量和度数之间的关系。在等奇异性理论中，人们寻求基于光纤多重性的标准，使一组解析空间是惠特尼等奇异的，因此拓扑平凡。pi计划通过使用由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。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 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.