Frobenius singularities and related invariants

弗罗贝尼乌斯奇点和相关不变量

• 批准号: 1419448
• 负责人: Kevin Tucker
• 金额: $ 14.4万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-16 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1419448&HistoricalAwards=false
• 关键词: Frobenius; singularities; related; invariants

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.

由Frobenius自同态定义的奇点和数值不变量是正特征交换代数和代数几何研究的重要内容。 为此，PI提出的计划将专注于F签名和其他相关的所谓F不变量，包括希尔伯特-昆兹多重性和测试理想。中央调查的PI是互动的几何方法在特征零源于复杂的代数几何。该计划的主要目标之一是更好地描述F签名的几何形状和更广泛的连接，以及它的许多推广。 PI旨在解决与该领域中一些长期存在的开放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和项猜想。 此外，PI计划建立在最近的工作描述测试理想，通过定期更改在探索局部和全局几何的代数簇的积极characteristic.Commutative代数和代数几何是最古老的，但最活跃的学科在数学。这些领域与复杂分析，拓扑学和数论等不同领域有着密切的联系，并用于各种各样的应用环境。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。 这些领域寻求理解几何对象（代数簇）局部给定的多项式方程的解决方案。例如，平面曲线是二元多项式的零点集（如尖点y^2 = x^3）。多项式方程的丰富性和简单性使代数簇成为令人着迷的研究对象。 PI建议研究的特定问题将有望导致对正特征中的多样性和奇异性的更深入理解，即具有素数消失属性的数系统。 特别地，这些系统包括基本上所有电子计算的核心的有限域。