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学，这是数学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。虽然代数几何形状的起源可以追溯到欧几里得和毕达哥拉斯的作品，但如今代数几何的重点通常是奇异性，这与邻居相比是不寻常的。例如，在一个图中，越过的点是曲线上唯一同类的点 - 一种奇异性。奇异性在自然界中经常出现，因为漏斗云的尖端，黑洞或物理状态的突然变化，例如球从墙壁弹起。它们通常表示给定物理系统异常或不稳定的状态。该项目涉及培训研究和教育活动的研究生和本科生，从而将研究与下一代的科学进步联系起来。该项目在D-Modules上有三个部分。第一部分的中心是d-sodules，它们来自奇异变化中的嵌入中。这里的较大目标包括研究这些模块上的Hodge理论过滤及其与相交同源性的相互作用。为了构想Hellus的猜想，该项目还将旨在建立某种消失的结果。第二部分始于特定类型的D模块，该模块在存在对品种的情况下出现。对于由此产生的等效d模块，在先前的NSF资助的工作中开发了PI，在复曲面的情况下进行了分类范式研究它们。将这些工具调整为Yau等人的重言式系统研究案例。在《镜像中》中，对称是项目的这一方面的核心。为了调查其功能演示，目标包括查找模量空间和周期模块的描述。要研究的第三种不变式是奇异性的伯恩斯坦 - 莎托多项式，这与溶液通过广泛的单片猜想密切相关。基于翻译（通过有限特征的Frobenius阈值）引入了一种意外的方法，以实现超几何D模块的新功能。在所有这些部分中，PI将指导论文项目并参与研究。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛的影响审查标准来评估而被视为珍贵的支持。