Homological approaches to differential forms, differential operators, and transfer of algebra structures

微分形式、微分算子和代数结构传递的同调方法

• 批准号: 2302198
• 负责人: Claudia Miller
• 金额: $ 16.5万
• 依托单位: Syracuse University
• 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: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302198&HistoricalAwards=false
• 关键词: Homological; approaches; differential; forms; operators

One of the main questions of algebraic geometry is: How does one find measures of singularity? A singularity is a place on a curve, surface, or higher dimensional space where it is not smooth, i.e., it has a sharp point or crosses itself. In general, one wants to be able to find measures of how extreme a singularity is and how these vary under maps, but usually the number of equations or variables makes a graph impossible or misleading. This is solved by employing the structural backbone given by algebraic geometry and commutative algebra. Generally, the goal is to attach invariants to singularities that are measures of the character of the singularity. Singularities will be studied in several ways: via differential forms and differential operators, which characterize smoothness; and by constructing new algebraic structures which yield not only invariants but also tools to discover new invariants. In the long run, work on singularities can lead to applications to computer vision and medical imaging and to string theory in physics. The project will continue the PI’s strong engagement in graduate education with PhD students. The PI will continue to co-organize conferences and workshops such as the Introductory Workshop at an MSRI semester program in 2024 and a future MSRI Summer Graduate School at the Chern Institute. Commutative and homological algebra are crucial in developing the foundations of algebraic geometry. This project will address some central topics and their interplay: differential forms, differential operators, cotangent complexes, and dg-algebra and A-infinity structures. The focus will be on (1) resolutions of differential forms and the cotangent complex, (2) differential operators of fixed orders and their resolutions, (3) DG algebra resolutions of graded Artinian algebras, and (4) comparisons of bar and Eagon resolutions via A-infinity structures. For the first, the focus is on understanding symmetries in and vanishing of invariants obtained from higher differentials, such as generalized Tjurina numbers, by showing how their resolutions are interrelated for Gorenstein singularities. For the second, a new homological approach will be used to work on finding the differential operators of hypersurfaces. The third involves the use of HPT (homological perturbation theory) to transfer algebra structures, and the fourth use of A-infinity structures to relate and generalize two classical resolutions. The unifying theme is to use homological methods to gain insight into these problems. The central challenge is to understand the homological behavior of exterior and symmetric power operations, which is known to be more involved and less understood, despite the fact that exterior algebras and power operations play a central role throughout many parts of algebra and geometry.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.

代数几何的一个主要问题是：如何找到奇点的度量？奇点是指曲线、曲面或高维空间上不光滑的地方，也就是说，它有一个尖锐的点或交叉。一般来说，人们希望能够找到奇点有多极端的测量方法，以及这些在地图上的变化情况，但通常情况下，方程或变量的数量使图形不可能或具有误导性。这是通过利用代数几何和交换代数给出的结构主干来解决的。一般来说，目标是将不变量附加到奇点，这些奇点是奇点特征的度量。奇点将通过几种方式进行研究：通过表征光滑性的微分形式和微分算子；通过构造新的代数结构，不仅产生不变量，而且还产生发现新不变量的工具。从长远来看，对奇点的研究可能会导致计算机视觉和医学成像的应用，以及物理学中的弦理论。该项目将继续推动研究生教育与博士生的紧密合作。国际和平研究所将继续共同组织会议和讲习班，如2024年MSRI学期方案的介绍性讲习班，以及未来在陈氏研究所的MSRI暑期研究生院。交换代数和同调代数在发展代数几何的基础方面是至关重要的。这个项目将解决一些中心主题及其相互作用：微分形式，微分算子，余切复形，dg-代数和A-无穷结构。重点将集中在(1)微分形式和余切复的分解，(2)定阶微分算子及其分解，(3)分次Artin代数的DG代数分解，以及(4)通过A-无穷结构比较BAR和Eagon分解。首先，重点是通过展示它们的分辨率如何与Gorenstein奇点相关来理解从高阶微分获得的不变量的对称性以及不变量的消失，例如广义Tjuina数。对于第二种方法，将使用一种新的同调方法来寻找超曲面的微分算子。第三种是使用HPT(同调微扰理论)来转移代数结构，第四种是使用A-无穷结构来联系和推广两个经典的分解。统一的主题是使用同源方法来洞察这些问题。中心挑战是理解外部和对称幂运算的同调行为，尽管外部代数和幂运算在代数和几何的许多部分都扮演着核心角色，但众所周知，这一行为更复杂，了解更少。该奖项反映了NSF的法定使命，并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，被认为值得支持。