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

协作研究：以等奇性理论为视角的微分方法、隐式化和多重性

• 批准号: 2201149
• 负责人: Bernd Ulrich
• 金额: $ 20.5万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-15 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2201149&HistoricalAwards=false
• 关键词: Collaborative; Research; Differential; Methods; Implicitization

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.

该研究项目涉及几何对象的结构，这些几何对象是多个变量中多项式平衡系统的解决方案集。这种称为品种的物体在整个数学以及科学与工程中的应用中都起着基本作用。研究人员将于1891年以数学家亨利·波卡（HenriPoincaré）为动机，研究人员将研究各种位置上各种切线的当地属性与各种切线的全球特征之间的关系，因为这些观点各不相同。他们还将致力于含义，这是纯数学中的一个经典问题，这对科学家在几何建模和计算机辅助设计方面引起了人们的关注。给定任何几何对象（例如曲线或表面），目标是找到具有几何对象作为解决方案集的多项式方程式的系统；了解这些“隐式”方程提供了对几何对象的见解。 PI将参与该项目的研究生和博士后访问者，它们将通过组织国际计划，会议和国家在线半决赛来促进科学交流。研究人员将研究与代数矢量领域有关的项目，与代数矢量有关的项目，REES环，等价理论和残留分裂的含义。 PI将使用交通证代数中的工具来研究如何在与该品种相切的矢量场的属性中反映出多种多样的全局不变性的类型。特别是，他们希望一方面建立射影平面曲线的奇异性的类型和星座与另一只手的分级Betti数字在另一方面的衍生模块。确定定义射击空间之间有理图的图形和图像的隐式方程是消除理论中的经典问题。 PI将集中在Cremona Maps的情况下，其中图的隐式方程还提供了反映射的参数。对于主要的有理图，他们将研究有理图的投射程度与定义图形的股票的数字和二元格之间的关系。在等式理论中，一个人寻求基于光纤的基于多样性的标准，以使分析空间家族成为惠特尼同等的，因此在拓扑上是琐碎的。 PIS计划通过使用由相交理论启发的新多重性通知来设计具有任意奇点的分析空间的标准。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛的影响审查标准来表示支持的支持。