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.

拟议的研究旨在研究交换代数与经典代数几何形状之间的微妙相互作用，另一方面表示理论和模块化算术。探索的主要主题是同源性，可以将其视为（DIS）相似性的研究。尽管被认为是一种抽象的数学工具，但同源性在近年来发现了许多应用程序，尤其是在数据分析和计算机科学方面。在代数中，同源性是指测量隐式定义的对象集（循环）和明确定义的子集（边界）之间的差异的一种方法。在几何形状中，它是一种用于区分不同形状的工具。附属物体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其对称性。 PI将研究与标志品种相关的同源理论，这些理论是几何对象，参数为线性空间子空间的增加序列，例如在平面中包含的线中包含的点。对称性来自移动线性空间，同时保留其遏制关系。所考虑的同源理论取决于质量数p = 2,3,5,7,11等，目标是了解产生的同源性如何取决于p，以及它如何编码标志品种的对称性。然后，这可以进一步应用于研究感兴趣的代数几何对象，例如矩阵和较高的张量。拟议的研究适合于吸引学生参与研究，以及计算机实验和软件开发。在交换代数，代数几何形状和代表理论的基本问题上，描述了Flag品种的线条捆绑关系。一个众所周知的情况是投射空间的情况，它等同于在变量理想中为多项式环计算局部同一个同时组。其他示例包括司羊grass骨，完整的国旗品种或发病率信函。对于特征零领域的国旗品种，同时理学是由Borel-Weil-Bott Theorem计算的，但是积极的特征问题仍然敞开。 PI的目标是对这个问题的系统研究，重点是具体示例，并展示了代数，几何和对称性之间的特殊相互作用。 PI计划利用新获得的知识来解决有关基本对象的问题，例如矩阵确定品种和高等张量，方程式和Syzygies或Koszul模块的类似物，或Koszul模块。该奖项反映了NSF的法定任务，并通过评估了基金会的范围，并通过评估了基金会的范围。