Interaction of Commutative Algebra, Valuations, and Geometry

交换代数、估值和几何的相互作用

基本信息

批准号：
2054394
负责人：
Steven Cutkosky
金额：
$ 31.64万
依托单位：
University of Missouri-Columbia
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054394&HistoricalAwards=false
关键词：
Interaction Commutative Algebra Valuations Geometry

项目摘要

This project is on the interaction of algebra and geometry. A major focus is on the analysis of singularities and their resolution. Resolution of singularities is the process of smoothing out, by algebraic operations, corners and cusps in a space defined by polynomial equations. This is of importance throughout mathematics and physics and has potential application to engineering. One direct application of research from this project is to the implicitation problem in computer aided geometric design. Commutative Algebra, Valuation Theory, Algebraic Geometry and Convex Geometry are unified in this project. An important focus of the project will be the training of graduate students and the mentoring of young mathematicians from diverse backgrounds. The theory of multiplicities and mixed multiplicities will be developed, by extending the theory from filtrations of m-primary ideals to arbitrary filtrations. Within this theory, the question of upper semicontinuity of multiplicity will be studied. The methods of the convex geometry of Okounkov bodies, commutative algebra and valuation theory will be fundamental in this project. The problem of determining a largest filtration which contains a given filtration and has the same multiplicity will be investigated. The theory of mixed multiplicities of line bundles on a projective variety will be studied, with the goal of characterizing when the Minkowski equality holds. Local uniformization of Abhyankar valuations dominating arbitrary excellent local rings, and the role of defect in local uniformization will be investigated.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.

该项目是关于代数和几何的相互作用。主要重点是奇点的分析及其解决。奇点的解决是通过代数运算平滑由多项式方程定义的空间中的角点和尖点的过程。这在整个数学和物理学中都很重要，并且在工程中具有潜在的应用。该项目研究的一个直接应用是计算机辅助几何设计中的隐含问题。交换代数、估价理论、代数几何和凸几何在这个项目中得到了统一。该项目的一个重要重点将是研究生的培训和来自不同背景的年轻数学家的指导。通过将理论从 m 次理想的过滤扩展到任意过滤，多重性和混合多重性理论将得到发展。在该理论中，将研究多重性的上半连续性问题。奥孔科夫体的凸几何方法、交换代数和估值理论将是该项目的基础。将研究确定包含给定过滤并具有相同多重性的最大过滤的问题。将研究射影簇上线束的混合多重性理论，目的是表征闵可夫斯基等式何时成立。 Abhyankar 估值的局部统一主导任意优秀局部环，以及局部统一中缺陷的作用将被调查。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。