Frobenius singularities and related invariants

由Frobenius自同态定义的奇点和数值不变量是交换代数和正特征代数几何研究的重要内容。为此，PI提出的方案将重点研究F-签名和其他相关的所谓F-不变量，包括希尔伯特-昆兹重数和测试理想。PI研究的中心是从复代数几何出发的特征零点与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-签名的几何和更广泛的联系，以及它的许多推广。PI的目的是解决与该领域一些长期悬而未决的问题有关的各种问题，包括弱F-正则性与强F-正则性的等价性以及直接求和猜想。此外，PI计划在最近描述测试理想的工作的基础上，通过规则的改变来探索正特征的代数簇的局部和全局几何。交换代数和代数几何是数学中最古老和最活跃的学科之一。这些领域与复杂分析、拓扑学和数论等不同领域有很强的联系，并在广泛的应用环境中使用。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。这些领域寻求理解本地给出的几何对象(代数簇)作为多项式方程的解。例如，平面曲线是两个变量的多项式的零集(如尖点y^2=x^3)。多项式方程的丰富性和简单性使代数变体成为吸引人的研究对象。PI建议研究的具体问题将有望导致对正特征中的变体和奇点的更深层次的理解，即具有素数为零的性质的超数字系统。具体地说，这些系统包括基本上所有电子计算的核心有限域。

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.