Frobenius singularities and related invariants

由Frobenius自同态定义的奇异点和数值不变量是交换代数和代数几何中正特征研究的重要组成部分。为此，PI提出的计划将重点关注f签名和其他相关的f不变量，包括Hilbert-Kunz多重性和测试理想。PI研究的核心是与源自复杂代数几何的特征零的几何方法的相互作用。该程序的主要目标之一是更好地描述f特征的几何形状和更广泛的联系，以及它的许多推广。PI旨在解决与该领域中许多长期存在的开放性问题相关的各种问题，包括弱与强f正则性的等价性和直接求和猜想。此外，PI计划在最近的工作基础上，通过探索正特征代数变体的局部和全局几何的定期变化来描述测试理想。交换代数和代数几何是数学中最古老也是最活跃的学科。这些领域与复杂分析、拓扑和数论等不同领域有着密切的联系，并被广泛应用于各种应用环境中。应用范围从计算机科学和基因组学中的纠错码到工程中的控制理论和建模。这些领域试图理解局部给定的几何对象（代数变量）作为多项式方程的解。例如，平面曲线是两个变量多项式的零集（如y^2 = x^3）。多项式方程的丰富性和简洁性使代数变量成为令人着迷的研究对象。PI提出研究的特定问题将有望导致对正特征（即具有素数消失性质的数系统）的变异和奇异性的更深入的理解。特别是，这些系统将有限场作为所有电子计算的核心。

Singularities and numerical invariants defined via the Frobenius endomorphism are an important part of the study of Commutative Algebra and Algebraic Geometry in positive characteristic. To that end, the program proposed by the PI will focus on the F-signature and other related so-called F-invariants, including Hilbert-Kunz multiplicity and test ideals. Central to the investigations of the PI is the interaction with geometric methods in characteristic zero stemming from complex algebraic geometry. One of the main objectives of the program is to better describe the geometry and broader connections of F-signature, as well as its many generalizations. The PI aims to approach various problems related to a number of long standing open questions in the field, including the equivalence of weak versus strong F-regularity and the direct summand conjecture. Furthermore, the PI plans to build upon recent work describing test ideals via regular alterations in exploring the local and global geometry of algebraic varieties in positive characteristic.Commutative Algebra and Algebraic Geometry are among the oldest and yet most active disciplines in mathematics. The fields have strong ties to such diverse areas as complex analysis, topology, and number theory, and are used in a wide variety of applied settings. Applications range from error-correcting codes in computer science and genomics to control theory and modeling in engineering. These fields seek to understand geometric objects (algebraic varieties) given locally as the solutions to polynomial equations. For instance, a plane curve is the zero set of a polynomial in two variables (such as the cusp y^2 = x^3). The richness and simplicity of polynomial equations make algebraic varieties fascinating objects of study. The particular questions the PI proposes to study will hopefully lead to a deeper understanding of the varieties and singularities in positive characteristic, i.e. over number systems having the property that a prime number vanishes. In particular, these systems include the finite fields at the heart of essentially all electronic computation.