D-Modules and Commutative Algebra

这个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，如机器人学、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，但今天代数几何的焦点往往是奇点，与它们的邻居相比，奇点是不寻常的。例如，在8字形中，交叉点是曲线上唯一的交叉点--奇点。奇点经常出现在自然界中，比如漏斗云的尖端，在黑洞中，或者在物理状态的突然变化中，比如球从墙上反弹。它们通常表示给定物理系统变得异常或不稳定的状态。这个项目包括对研究生和本科生的研究和教育活动的培训，从而将研究与下一代的科学进步联系起来。这个项目包括三个部分，关于D模块。第一部分的中心是D-模，它是由奇异簇嵌入流形而产生的。这里更大的目标包括研究这些模上的Hodge理论滤子以及它们与交同调的相互作用。考虑到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-模开始，这种D-模是在对一个品种的集体行动中出现的。对于所得到的等变D-模，PI在之前的NSF资助的工作中开发了一个范畴范式来研究它们在环状情况下的情况。使这些工具适用于Yau等人研究的重言式系统的更一般情况。镜像对称是该项目这一方面的核心。为了研究它们的函数式表示，目标包括找到模空间和周期模的描述。第三种要研究的不变量是奇点的Bernstein-Sato多项式，它通过开单点猜想与解数密切相关。基于有限特征的Frobenius阈值到超几何D-模的一个新特征的转换，引入了对这一猜想的一种意想不到的方法。在所有这些部分，PI将指导论文项目并让本科生参与研究。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.