Measuring singularities in commutative algebra

这个项目涉及交换代数理论中的问题。这是一个处理代数变量的局部性质的领域，即，多变量多项式方程组的解集。例如，两个变量的单多项式方程的解集可以实现为平面上的曲线（如抛物线y=x^2）。代数类的奇异点或非光滑点具有丰富的代数和几何结构，详细描述它们的性质在许多研究中是至关重要的。例如，抛物线是非奇异的（这意味着它局部看起来像一条线），而由y^2=x^3定义的曲线除了在原点以外是非奇异的，在原点局部看起来像一个尖点。将探讨的项目集中在代数变种（即奇点）的奇点上，重点是使用各种代数技术测量奇点。该项目将由他的研究生和博士后参与。PI将与该领域的其他专家合作构建混合特征奇点理论。一个重点是开发测试理想的混合特征版本。项目包括研究它们在局部化和完成下的行为，以及它们与几何乘数理想的联系。另一个重点是使用bhat - scholze的完美化函子定义的完美样特征和完美样Hilbert-Kunz多重性，它们的灵感来自f -特征和来自正特征的Hilbert-Kunz多重性。项目包括理解这些数值不变量在局部化和家族中的行为。PI还将继续研究希尔伯特-塞缪尔多样性，重点关注长期存在的莱赫猜想。提出的研究项目包括研究三维混合特征情况，以及在分级情况之外探索lim Ulrich序列的存在性。PI还将研究lim Cohen-Macaulay序列及其变体的存在性，以攻击长期存在的Serre关于交集多重性的猜想。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 involves questions in the theory of commutative algebra. This is a field that deals with the local properties of algebraic varieties, i.e., the solution set of a system of polynomial equations in several variables. For example, the solution set of a single polynomial equation in two variables can be realized as a curve in the plane (e.g., a parabola y=x^2). The singular or non-smooth points of an algebraic variety have rich algebraic and geometric structures, and detailing their properties is crucial in many investigations. For example, the parabola is nonsingular (meaning that locally it looks like a line), while the curve defined by y^2=x^3 is nonsingular except at the origin, where it locally looks like a cusp. The projects that will be explored are focused on the singular points of algebraic varieties (i.e., singularities), with a focus on measuring singularities using various algebraic techniques. The PI will involve his graduate students and post-docs in this research project.The PI will construct a mixed characteristic singularity theory in collaboration with other experts in this area. One focus is to develop a mixed characteristic version of test ideals. Projects include studying their behaviors under localization and completion, and their connections to multiplier ideals from birational geometry. Another focus is on perfectoid signature and perfectoid Hilbert-Kunz multiplicity, which are defined using the perfectoidization functor of Bhatt-Scholze and are inspired by F-signature and Hilbert-Kunz multiplicity from positive characteristic. Projects include understanding the behavior of these numerical invariants under localization and in families. The PI will also continue the study on Hilbert-Samuel multiplicities, with a focus on the longstanding Lech's conjecture. The proposed projects include attacking the three dimensional mixed characteristic case, and exploring the existence of lim Ulrich sequences beyond the graded case. The PI will also investigate the existence of lim Cohen-Macaulay sequences and their variations to attack the longstanding Serre's conjecture on intersection multiplicities.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.