Singularities and Multiplicities in Commutative Algebra

PI计划研究交换代数的问题。这是一个研究代数变量的领域，也就是多变量多项式方程组的解集。例如，由方程y=x^2定义的抛物线在平面上是光滑的，这意味着在局部，在任何一点上，它看起来像一条直线，而由方程y^2=x^3定义的曲线在原点处不光滑（它看起来像原点附近的一个尖点）。代数类的奇异点或非光滑点具有非常丰富的代数和几何结构，研究它们的性质在其他科学和工程中有许多应用。将要探索的项目是对代数变量的奇点（即奇点）的研究，重点是使用理论乘数/测试理想和多重性来测量奇点（例如，分配一些不变量来相互比较）。PI计划开发一种与最小模型程序中的奇点平行的混合特征奇点理论和等特征紧闭理论。这是由伊夫·安德烈最近的突破所激发的，他解决了霍赫斯特长期存在的直接求和猜想和大科恩-麦考利代数的存在。与Karl Schwede一起，PI利用Andre的技术从完美代数和空间开始建立这样一个理论，引入了一个完美的乘数/测试理想版本。将应用于混合特征符号幂的研究，以及特征变化时奇点族的研究。另一个研究方向是奇点的Hilbert-Samuel多重性，重点是Lech猜想、Lech不等式和Stuckrad—Vogel不变量。PI解决了长期存在的Lech猜想的三维等特征情况，最近解决了Stuckrad- Vogel猜想（与Klein， Pham， Smirnov和Yao一起）。提出的项目包括通过研究和改进经典的莱赫不等式来攻击莱赫猜想的高维情况。一个相关的项目是研究足够深的理想的多样性和长度之间的关系。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。

The PI plans to study questions in commutative algebra. This is a field that deals with the study of algebraic varieties, that is, the solution set of a system of polynomial equations in several variables. For example, the parabola defined by the equation y=x^2 in the plane is smooth, meaning that locally, at any point, it looks like a line, while the curve defined by the equation y^2=x^3 is not smooth at the origin (it looks like a cusp near the origin). The singular or non-smooth points of an algebraic variety have very rich algebraic and geometric structures, and investigating their properties has many applications in other sciences and engineering. The projects that will be explored are the study of singular points of algebraic varieties (i.e., singularities), with a focus on measuring the singularities (e.g., assigning some invariant to compare them with each other) using the theory multiplier/test ideals and multiplicities. The PI plans to develop a mixed characteristic singularity theory that is parallel to singularities in the minimal model program and tight closure theory from equal characteristic. This is motivated by the recent breakthrough of Yves Andre that solved Hochster's long-standing direct summand conjecture and the existence of big Cohen-Macaulay algebras. Together with Karl Schwede, the PI utilizes Andre`'s techniques from perfectoid algebras and spaces to start building such a theory, introducing a perfectoid version of multiplier/test ideals. There will be applications on the study of symbolic powers in mixed characteristic, as well as the study of family of singularities when the characteristic varies. Another research direction is the Hilbert-Samuel multiplicities attached to singularities, with a focus on Lech's conjecture, Lech's inequality, and the Stuckrad--Vogel invariant. The PI has settled the dimension three equal characteristic case of the longstanding Lech's conjecture, and has recently settled the Stuckrad--Vogel conjecture (together with Klein, Pham, Smirnov and Yao). The proposed projects include attacking the higher dimensional case of Lech's conjecture by studying and improving the classical Lech's inequality. A related project is to investigate the relationship between multiplicity and colength for sufficiently deep ideals.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.