Homological Commutative Algebra and Symmetry

同调交换代数和对称性

• 批准号: 2302341
• 负责人: Claudiu Raicu
• 金额: $ 35万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302341&HistoricalAwards=false
• 关键词: Homological; Commutative; Algebra; Symmetry

The proposed research aims to investigate subtle interactions between commutative algebra and classical algebraic geometry on the one hand, and representation theory and modular arithmetic on the other hand. The main subject of exploration is homology, which can be thought of as the study of (dis)similarities. Despite being regarded as an abstract mathematical tool, homology has found many applications in recent years, particularly to data analysis and computer science. In algebra, homology refers to a way of measuring the difference between an implicitly defined set of objects (cycles) and an explicitly defined subset (of boundaries). In geometry, it is a tool used to distinguish between different shapes. The homology attached to an object usually focuses on the most significant traits, and it reflects its symmetries in intriguing ways. The PI will investigate homological theories associated to flag varieties, which are geometric objects parametrizing increasing sequences of subspaces of linear spaces, such as a point contained in a line contained in a plane. The symmetry comes from moving the linear spaces around while preserving their containment relations. The homological theories considered depend on a prime number p=2,3,5,7,11 etc., and the goal is to understand how the resulting homology depends on p, and how it encodes the symmetries of the flag varieties. This can then be further applied to study algebro-geometric objects of interest, such as matrices and higher-dimensional tensors. The proposed research is suitable for engaging students in research, as well as for computer experimentation and software development.A fundamental question at the confluence of commutative algebra, algebraic geometry and representation theory is to describe the cohomology of line bundles on flag varieties. A well-known case is that of the projective space, where it is equivalent to computing local cohomology for a polynomial ring with support in the ideal of the variables. Other examples include Grassmannians, complete flag varieties, or the incidence correspondence. For flag varieties over a field of characteristic zero the cohomology is computed by the Borel-Weil-Bott theorem, but the positive characteristic problem remains wide open. The PI's goal is a systematic study of this question, with an emphasis on concrete examples, and on showcasing peculiar interactions between algebra, geometry, and symmetry. The PI plans to use the newly acquired knowledge to solve questions of a homological nature regarding fundamental objects such as matrix determinantal varieties and analogues for higher tensors, equations and syzygies, or Koszul modules.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.

本研究旨在探讨交换代数与经典代数几何之间的微妙相互作用，以及表示理论与模算法之间的微妙相互作用。探索的主要主题是同源性，这可以被认为是对（非）相似性的研究。尽管被认为是一种抽象的数学工具，但近年来，同源性已经发现了许多应用，特别是在数据分析和计算机科学中。在代数中，同调是指一种测量隐式定义的对象集合（圈）和显式定义的子集（边界）之间差异的方法。在几何学中，它是用来区分不同形状的工具。附在一个物体上的同源性通常集中在最重要的特征上，它以有趣的方式反映了它的对称性。PI将研究与标志变体相关的同调理论，标志变体是参数化线性空间子空间的递增序列的几何对象，例如包含在平面中包含的直线中的点。对称来自于移动线性空间的同时保持它们的包容关系。所考虑的同调理论依赖于素数p=2,3,5,7,11等，目的是了解所得到的同调如何依赖于p，以及它如何编码旗变体的对称性。这可以进一步应用于研究感兴趣的代数几何对象，如矩阵和高维张量。建议的研究适合学生参与研究，以及计算机实验和软件开发。交换代数、代数几何和表示理论相互融合的一个基本问题是描述旗变上束的上同调性。一个众所周知的例子是在射影空间中，它等价于计算一个多项式环在变量的理想支持下的局部上同调。其他的例子包括格拉斯曼式、完整的标志变体或关联对应。在特征为零的域上，用Borel-Weil-Bott定理计算了旗变的上同性，但其正特征问题仍有待解决。PI的目标是对这个问题进行系统的研究，强调具体的例子，并展示代数、几何和对称之间的特殊相互作用。PI计划利用新获得的知识来解决关于基本对象的同调性质的问题，如矩阵行列式变化和高张量的类似物，方程和协同，或Koszul模块。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。