Homological Commutative Algebra and Symmetry

拟议的研究旨在调查交换代数和经典代数几何之间的微妙的相互作用，一方面，表示论和模块算术的另一方面。探索的主要主题是同源性，可以被认为是对（不）相似性的研究。尽管被认为是一个抽象的数学工具，同源性近年来已经发现了许多应用，特别是数据分析和计算机科学。在代数中，同调是指一种测量隐式定义的对象（循环）集合和显式定义的（边界）子集之间差异的方法。在几何学中，它是用来区分不同形状的工具。与一个物体相关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将研究与旗簇相关的同调理论，旗簇是对线性空间的子空间的递增序列进行参数化的几何对象，例如包含在平面中的线中的点。对称性来自于线性空间的移动，同时保持它们的包含关系。所考虑的同调理论依赖于素数p= 2，3，5，7，11等，我们的目标是了解所得到的同源性是如何依赖于p的，以及它是如何编码标志变体的对称性的。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。该研究适合于学生从事研究，以及计算机实验和软件开发。交换代数，代数几何和表示论交汇的一个基本问题是描述旗簇上的线丛的上同调。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，该多项式环的支撑在变量的理想中。其他例子包括格拉斯曼人，完整的旗变种，或发病率对应。对于特征为零的域上的旗簇，上同调由Borel-Weil-Bott定理计算，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，几何和对称之间的特殊相互作用。PI计划使用新获得的知识来解决有关基本对象的同调性质的问题，例如矩阵行列式变体和更高张量的类似物，方程和合朔，或Koszul模块。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.