Combinatorics, Cohomology, and Matrix Spaces

组合学是研究具体有限物体的数学领域。在过去的半个世纪里，它已经从一个不起眼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厚的联系和应用。从一开始就刺激组合学发展的核心数学框架是通过将几何空间分解成组成部分来理解几何空间的挑战。这种类型的最新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被称为矩阵空间的二维数据数组集合。该项目将使用组合工具来确定通过对底层数据施加冗余条件来确定分解成块的矩阵空间的属性。此外，资金将用于支持本科生研究、研究生培训和包括监狱教育项目在内的推广工作。对于一个数学家来说，这个项目研究的空间非常简单：矩阵、幂等矩阵、对称和偏对称矩阵。本文所考虑的矩阵空间块称为矩阵舒伯特变集，由满足特定子矩阵上秩限制的矩阵组成。矩阵舒伯特变在某些行/列操作下是不变的——这扩展到自然的群作用。考虑到这一群作用，矩阵舒伯特变的几何和拓扑性质解决了枚举代数几何和交理论中的核心问题。使用组合学和交换代数中的一系列工具，统称为“Grobner几何”，该项目将描述编码矩阵舒伯特变的几何性质的多项式表示。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。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。

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.