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.

奇点是几何物体以某种方式退化的点。它们在代数几何的研究中起着重要的作用。本项目涉及使用某些新技术来研究奇点的不变量。这一理论为研究线性偏微分方程提供了一种代数方法，但它在代数几何和相关学科中也有许多应用。这个项目的目标是，一方面，进一步研究一些最近引入的奇点类，另一方面，探索与zeta函数和不变量的新联系。该提案为PI与研究生和博士后合作提供了充分的机会。他还将就这个问题写一本研究专著。这个项目将涉及几个不同的研究方向。首先，PI将扩展k-有理和k-Du Bois奇点的研究，超越超曲面的情况，使用最近定义的局部完全相交的最小指数版本。在另一个方向上，他将研究给定素数的奇异模幂的d模理论不变量，从正特征中提炼检验理想和f纯阈值，并可能提供特征零不变量的算术对应，例如Hodge理想和最小指数。第三个方向是探索动机zeta函数和极小指数之间的联系，以及拓扑zeta函数和d -模块之间的联系（可能应用于强单性猜想）。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。