Combinatorics, Cohomology, and Matrix Spaces

组合学、上同调和矩阵空间

• 批准号: 2054423
• 负责人: Zachary Hamaker
• 金额: $ 29.98万
• 依托单位: University of Florida
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054423&HistoricalAwards=false
• 关键词: Combinatorics; Cohomology; Matrix; Spaces

Combinatorics is the field of mathematics most concerned with concrete, finite objects. Over the past half century, it has grown from humble origins into a rich field with deep connections to and applications in disparate fields across mathematics and the sciences. A central mathematical framework that has spurred combinatorics' development from its inception is the challenge of understanding geometric spaces by their decomposition into constituent pieces. Recent breakthroughs of this type have led to deeper understanding of fundamental problems in theoretical physics, probability theory, algebraic geometry and many other fields. Since the 19th century, mathematicians have applied this framework to better understand collections of two-dimensional arrays of data called matrix spaces. This project will use combinatorial tools to determine properties of matrix spaces broken down into pieces determined by imposing redundancy conditions on the underlying data. In addition, funds will support undergraduate research, training graduate students and outreach efforts including work with a prison education program.To a mathematician, the spaces this project studies are quite simple: matrices, idempotent matrices, symmetric and skew-symmetric matrices. The pieces of matrix spaces considered in this project, called matrix Schubert varieties, are comprised of matrices satisfying rank restrictions on specified submatrices. Matrix Schubert varieties are invariant under certain row/column operations -- this extends to a natural group action. By taking this group action into account, geometric and topological properties of matrix Schubert varieties solve central questions in enumerative algebraic geometry and intersection theory. Using a collection of tools from combinatorics and commutative algebra known collectively as 'Grobner geometry', this project will describe polynomial representatives that encode the geometric properties of matrix Schubert varieties. A potential application is the first combinatorial description of structure constants in the Lagrangian Grassmannian.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.

组合数学是最关注具体的、有限的对象的数学领域。在过去的半个世纪里，它已经从卑微的起源发展成为一个丰富的领域，与数学和科学的不同领域有着深刻的联系和应用。从一开始就推动组合学发展的一个核心数学框架是通过将几何空间分解为组成部分来理解几何空间的挑战。最近这类突破使人们对理论物理、概率论、代数几何和许多其他领域的基本问题有了更深入的理解。自19世纪以来，数学家们已经应用这个框架来更好地理解称为矩阵空间的二维数据数组的集合。这个项目将使用组合工具来确定矩阵空间的属性，通过对底层数据施加冗余条件来确定矩阵空间的属性。此外，基金将支持本科生研究，培训研究生和推广工作，包括与监狱教育计划的工作。对于数学家来说，这个项目研究的空间非常简单：矩阵，幂等矩阵，对称和反对称矩阵。在这个项目中考虑的矩阵空间，称为矩阵舒伯特簇，由满足特定子矩阵秩限制的矩阵组成。矩阵舒伯特簇在某些行/列操作下是不变的-这扩展到自然的群作用。通过考虑这种群作用，矩阵舒伯特簇的几何和拓扑性质解决了枚举代数几何和交集理论中的中心问题。使用集合的工具，从组合学和交换代数统称为'Grobner几何'，这个项目将描述多项式的代表，编码矩阵舒伯特品种的几何属性。一个潜在的应用是拉格朗日格拉斯曼结构常数的第一个组合描述。这个奖项反映了NSF的法定使命，并已被认为是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的支持。

项目成果

Involutions under Bruhat order and labeled Motzkin paths

Bruhat 阶下的对合和标记的 Motzkin 路径

• DOI: 10.1016/j.ejc.2022.103513
• 发表时间: 2022
• 期刊: European Journal of Combinatorics
• 影响因子: 1
• 作者: Coopman, Michael;Hamaker, Zachary
• 通讯作者: Hamaker, Zachary

Lenart's Bijection via Bumpless Pipe Dreams

莱纳特通过无扰动白日梦实现的双射

• DOI: 10.37236/11391
• 发表时间: 2023
• 期刊: The Electronic Journal of Combinatorics
• 作者: Gregory, Adam;Hamaker, Zachary
• 通讯作者: Hamaker, Zachary