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D-Modules and Commutative Algebra

D 模和交换代数

基本信息

批准号：
2100288
负责人：
Hans Ulrich Walther
金额：
$ 35万
依托单位：
Purdue University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2100288&HistoricalAwards=false
关键词：
Modules Commutative Algebra

项目摘要

The focus of this project is in algebraic geometry, one of the most varied areas of mathematics, permeating many different branches of science such as robotics, cosmology, and computer encryption. While the origins of algebraic geometry can be traced to the works of Euclid and Pythagoras, the focus of algebraic geometry today is often on singularities, which are points that are unusual when compared to their neighbors. For example, in a figure-eight, the point of crossing is the only one of its kind on the curve – a singularity. Singularities appear frequently in nature, as the tip of a funnel cloud, in a black hole, or in abrupt changes of physical states such as a ball bouncing off a wall. They typically indicate states in which a given physical system becomes anomalous or unstable. This project involves training of graduate and undergraduate students in research and educational activities, thus connecting research with the scientific advancement of the next generation.There are three parts to this project on D-modules. At the center of the first part are D-modules that arise from embeddings of singular varieties into manifolds. The larger goals here include studying Hodge theoretic filtrations on these modules and their interplay with intersection homology. With a view towards a conjecture of Hellus, the project will also aim at establishing a certain type of vanishing result. The second piece begins with a particular type of D-module that arises in the presence of a group action on a variety. For the resulting equivariant D-modules, the PI developed in previous NSF-funded work a categorical paradigm to study them in the toric case. Adapting these tools to the more general case of tautological systems studied by Yau et al. in mirror symmetry is at the core of this aspect of the project. To investigate their functorial presentations, goals include finding a description of the moduli space and the period module. A third type of invariant to be investigated is the Bernstein–Sato polynomial of a singularity, which is closely related to solution counts via the wide-open Monodromy Conjecture. An unexpected approach to this conjecture is introduced, based on a translation – via Frobenius thresholds in finite characteristic – to a new feature of hypergeometric D-modules. In all these parts, the PI will be directing thesis projects and engage undergraduates in research.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.

该项目的重点是代数几何，这是数学中最多样化的领域之一，渗透到许多不同的科学分支，例如机器人技术、宇宙学和计算机加密。虽然代数几何的起源可以追溯到欧几里得和毕达哥拉斯的作品，但今天代数几何的重点往往是奇点，奇点是与其邻居相比不寻常的点。例如，在一个8字形曲线中，交点是曲线上唯一的一个交点--奇点。奇点在自然界中经常出现，如漏斗云的尖端，黑洞中，或物理状态的突然变化，如球从墙上反弹。它们通常表示给定物理系统变得异常或不稳定的状态。该项目涉及在研究和教育活动中培训研究生和本科生，从而将研究与下一代的科学进步联系起来。在第一部分的中心是D-模，它产生于奇异簇嵌入流形。这里更大的目标包括研究霍奇理论过滤这些模块和它们的相互作用与交叉同源性。为了实现Hellus的猜想，该项目还将致力于建立某种类型的消失结果。第二部分从一种特殊类型的D-模块开始，这种模块是在一个群体对一个品种的作用下产生的。对于由此产生的同变D-模，PI在以前的NSF资助的工作中开发了一个分类范式来研究它们在复曲面的情况下。使这些工具适用于Yau等人在镜像对称中研究的重言式系统的更一般情况是该项目这一方面的核心。为了研究它们的函子表示，目标包括找到模空间和周期模的描述。第三种类型的不变量是Bernstein-Sato多项式的奇异性，这是密切相关的解决方案计数通过广泛开放的单值猜想。一个意想不到的方法，这一猜想的基础上的翻译-通过Frobenius阈值在有限的特征-超几何D-模的一个新的功能。在所有这些部分中，PI将指导论文项目并吸引本科生参与研究。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。