D-modules and invariants of singularities

D 模和奇点不变量

• 批准号: 2301463
• 负责人: Mircea Mustata
• 金额: $ 35万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2301463&HistoricalAwards=false
• 关键词: modules; invariants; singularities

Singularities are points where a geometric object degenerates in some way. They play an important role in the study of algebraic geometry. This project is concerned with the use of certain novel techniques to study invariants of singularities. This theory provides an algebraic approach to the study of linear partial differential equations, but it has found many applications in algebraic geometry and related subjects. The goal of this project is, on one hand, to further study some recently introduced classes of singularities and, on the other hand, to explore new connections with zeta functions and invariants. The proposal provides ample opportunity for the PI to collaborate with graduate students and post-docs. He will also write a research monograph on this subject.This project will involve several different research directions. First, the PI will expand the study of k-rational and k-Du Bois singularities beyond the case of hypersurfaces, using the recently defined version of minimal exponent for locally complete intersections. In a different direction, he will investigate D-module theoretic invariants of singularities modulo powers of a given prime, that refine the test ideals and F-pure thresholds from positive characteristic, and that might provide an arithmetic counterpart to characteristic zero invariants, such as Hodge ideals and minimal exponents. A third direction concerns exploring connections between the motivic zeta function and the minimal exponent on one side, and between the topological zeta function and D-modules on the other side (with possible applications to the Strong Monodromy Conjecture).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.

奇点是几何对象以某种方式退化的点。它们在代数几何学的研究中起着重要作用。该项目涉及使用某些新型技术来研究奇异性的不变性。该理论为线性偏微分方程的研究提供了一种代数方法，但是它在代数几何和相关主题中发现了许多应用。该项目的目的是进一步研究一些最近引入的奇异性类别，另一方面，探索与Zeta功能和不变式的新连接。该提案为PI提供了足够的机会与研究生和研究生合作。他还将撰写有关此主题的研究专着。该项目将涉及几个不同的研究方向。首先，PI将使用最近定义的最小指数的最小指数来扩展Kational和K-Du Bois奇异性的研究。在不同的方向上，他将研究给定素数的奇异性奇异性的D模块理论不变性，从积极特征中完善测试理想和F-pure阈值，并且可能会提供与特征性零不变的算术相关性，例如hodge的理想和最小值。第三个方向涉及探索动机Zeta功能与一侧最小指数之间的联系，拓扑Zeta功能与另一侧的D模块之间（可能适用于强有力的单型猜想）。这项奖项反映了NSF的法定任务，并通过评估基础的Merit和Broadial and crarit和广泛的影响。