Singularities and Multiplicities in Commutative Algebra

PI计划研究交换代数中的问题。这是一个研究代数簇的领域，也就是说，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，在任何点，它看起来像一条线，而由方程y^2=x^3定义的曲线在原点不光滑（它看起来像原点附近的尖点）。代数簇的奇异点或非光滑点具有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。将探讨的项目是研究代数簇的奇点（即，奇点），重点在于测量奇点（例如，分配一些不变量以将它们彼此比较）使用理论乘数/测试理想和多重性。PI计划发展一种与最小模型程序中的奇异点和等特征紧闭理论并行的混合特征奇异点理论。这是由于最近Yves Andre的突破，解决了Hochster长期存在的直接和项猜想和大Cohen-Macaulay代数的存在性。与Karl Schwede一起，PI利用Andre的完美代数和空间技术开始构建这样的理论，引入了乘数/测试理想的完美版本。在混合特征的符号幂的研究以及特征变化时奇点族的研究中有应用。另一个研究方向是奇异性的希尔伯特-塞缪尔多重性，重点是莱赫猜想，莱赫不等式和Stuckrad-Vogel不变量。PI已经解决了长期存在的Lech猜想的三维相等特征情况，最近解决了Stuckrad-Vogel猜想（与Klein，Pham，Smirnov和Yao一起）。提出的方案包括通过研究和改进经典的Lech不等式来攻击高维情形的Lech猜想。一个相关的项目是调查多样性和颜色之间的关系，以达到足够深的理想。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的知识价值和更广泛的影响审查标准的支持。

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.