Computing and Interpreting Frobenius Invariants

交换代数和代数几何是数学中最古老也是最活跃的学科。这些领域与数学的各个领域有着密切的联系，也被用于各种各样的应用环境。本研究计划将探讨这些领域的重要开放问题，旨在更深入地了解正特征数系上簇的奇异性。这些系统，那些素数消失的系统，包括基本上所有电子计算核心的有限域。在开展这项研究计划的同时，研究员还将组织研究研讨会，支持和监督研究生的研究，并开展活动，包括年度研讨会，以促进本科生的研究。说明，研究人员计划继续研究的奇异性和不变量定义通过Frobenius映射在正特征交换代数。特别是，研究将集中在围绕F-签名，希尔伯特-昆兹多重性和测试理想，在两个非常不同的方向的想法圈。首先，在一个固定的特征，该项目的目的是扩大在计算这些不变量的最新进展，并探讨一些长期存在的重要的开放问题，如弱和强F-正则性的等价性。其次，研究的目的是表明这些不变量在还原为正特征时具有极限值。在这里，我们希望，极限的特点趋于无穷大有更简单的解释和价值，并可以与几何措施的奇异性的复代数簇。与几何方法的相互作用在特征零源于复杂的代数几何是该项目的核心，工作的主要目标之一是更好地描述几何和更广泛的F-不变量的连接。

Commutative algebra and algebraic geometry are among the oldest and yet most active disciplines in mathematics. The fields have strong ties to diverse areas of mathematics, and are also used in a wide variety of applied settings. This research project will explore important open questions in these fields, aiming for a deeper understanding of the singularities of varieties over positive characteristic number systems. These systems, those where a prime number vanishes, include the finite fields at the heart of essentially all electronic computation. While carrying out this research program, the investigator will also organize research seminars, support and supervise graduate student research, and conduct activities, including an annual symposium, to promote undergraduate research. Explicitly, the investigator plans to continue study of singularities of and invariants defined via the Frobenius map in positive characteristic commutative algebra. In particular, the research will focus on the circle of ideas surrounding F-signature, Hilbert-Kunz multiplicity, and test ideals, in two very different directions. First, in a fixed characteristic, the project aims to expand upon recent progress in computing these invariants, and to approach some long standing important open questions about them, such as the equivalence of weak and strong F-regularity. Second, the research aims to show these invariants have limiting values under reduction to positive characteristic. Here, it is hoped that the limits as the characteristic tends towards infinity have simpler interpretations and values, and can be related to geometric measures of singularities for complex algebraic varieties. The interaction with geometric methods in characteristic zero stemming from complex algebraic geometry is central to the project, and one of the main objectives of the work is to better describe the geometry and broader connections of F-invariants.