Measuring singularities in commutative algebra

这个项目涉及到交换代数理论中的问题。这是一个研究代数簇的局部性质的领域，即多变量多项式方程组的解集。例如，两个变量的单个多项式方程的解集可以实现为平面上的一条曲线(例如抛物线y=x^2)。代数族的奇点或非光滑点具有丰富的代数和几何结构，详细地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(意味着它在局部上看起来像一条直线)，而由y^2=x^3定义的曲线是非奇异的，除非在原点处，它在局部上看起来像一个尖点。将探索的项目集中在代数簇(即奇点)的奇点上，重点是使用各种代数技术来测量奇点。PI将邀请他的研究生和博士后参与这一研究项目，PI将与该领域的其他专家合作构建一个混合特征奇点理论。其中一个重点是开发一个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备化下的行为，以及它们与二次几何乘子理想的联系。另一个重点是由正特征得到的F-签名和Hilbert-kunz重数的启发，利用Bhatt-Scholze的完美化函子定义了完全拟签名和完全拟Hilbert-kunz重数。项目包括了解这些数值不变量在本地化和家族中的行为。PI还将继续对希尔伯特-塞缪尔多重性的研究，重点是长期存在的莱赫猜想。所提出的项目包括攻击三维混合特征情形，以及在分级情形之外探索Lim Ulrich序列的存在性。PI还将调查Lim Cohen-Macaulay序列及其变体的存在，以攻击长期存在的Serre关于交集倍数的猜想。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势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.