Interaction of Commutative Algebra, Valuations, and Geometry

这个项目是关于代数和几何的相互作用。一个主要的重点是奇点的分析和解决方案。奇点的解析是通过代数运算，在多项式方程定义的空间中平滑角点和尖点的过程。这在整个数学和物理学中都很重要，并在工程学中有潜在的应用。本计画之研究成果之一直接应用于电脑辅助几何设计中的隐含问题。本项目将交换代数、赋值理论、代数几何和凸几何统一起来。 该项目的一个重要重点将是培训研究生和指导来自不同背景的年轻数学家。多重性和混合多重性理论将通过将理论从m-准素理想的滤子扩展到任意滤子而得到发展。在这个理论中，将研究多重性的上连续性问题。奥昆科夫体的凸几何方法、交换代数和赋值理论将是本项目的基础。将研究确定包含给定滤子且具有相同重数的最大滤子的问题。 我们将研究射影簇上的线丛的混合重数理论，目标是刻画闵可夫斯基等式何时成立。本地均匀化的Abhyankar估值主导任意优秀的本地环，和本地均匀化的缺陷的作用将被调查。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.