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.

这项研究的目的是研究交换代数和经典代数几何之间的微妙相互作用，以及表示理论和模算术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(不)相似性的研究。尽管同调被认为是一种抽象的数学工具，但近年来它得到了许多应用，特别是在数据分析和计算机科学方面。在代数中，同调是指测量隐式定义的对象集(圈)和显式定义的子集(边界)之间的差异的一种方法。在几何学中，它是用于区分不同形状的工具。附着在物体上的同源性通常集中在最重要的特征上，它以耐人寻味的方式反映了它的对称性。PI将研究与旗帜变体相关的同调理论，旗帜变体是将线性空间的子空间的递增序列参数化的几何对象，例如平面中包含的直线中包含的点。对称性来自于移动线性空间，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，目标是理解由此产生的同调如何依赖于p，以及它如何编码旗簇的对称性。然后，这可以进一步应用于研究感兴趣的代数几何对象，例如矩阵和高维张量。该研究既适用于学生的研究，也适用于计算机实验和软件开发。在交换代数、代数几何和表示论的交汇处，一个基本问题是刻画旗簇上线丛的上同调。一个著名的例子是射影空间，它等价于计算一个多项式环的局部上同调，在变量理想中有支集。其他例子包括格拉斯曼尼亚人、完整的旗帜变种或发病率对应关系。对于特征零域上的FLAG簇，上同调是由Borel-Weil-Bott定理计算的，但正特征问题仍然是开放的。PI的目标是对这个问题进行系统的研究，重点是具体的例子，并展示代数、几何和对称性之间的特殊相互作用。PI计划使用新获得的知识来解决关于基本对象的同源性质的问题，如矩阵行列式变体和更高张量的类似物、方程和合子或Koszul模块。该奖项反映了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。