Interaction of Commutative Algebra, Valuations, and Geometry

这个项目是关于代数和几何的相互作用。一个主要的焦点是对奇点的分析和它们的解决。奇异点的解算是在多项式方程定义的空间中，通过代数运算，将角点和尖点平滑化的过程。这在整个数学和物理中都很重要，并且在工程中也有潜在的应用。本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，估值理论，代数几何和凸几何在这个项目中统一。该项目的一个重要重点将是培养研究生和指导来自不同背景的年轻数学家。通过将理论从m-初级理想的过滤扩展到任意过滤，将发展多重性和混合多重性理论。在这个理论中，我们将研究多重性的上半连续性问题。Okounkov体的凸几何方法，交换代数和估值理论将是本项目的基础。本文将研究包含给定过滤且具有相同多重度的最大过滤的确定问题。本文将研究线束在射影变化上的混合多重性理论，目的在于描述闵可夫斯基等式何时成立。研究了支配任意优局部环的Abhyankar值的局部均匀化，以及缺陷在局部均匀化中的作用。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 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.