The Algebra, Blueprinted Geometry, and Combinatorics of Matroids

拟阵的代数、蓝图几何和组合学

• 批准号: 2154224
• 负责人: Matthew Baker
• 金额: $ 30万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-10-01 至 2025-09-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154224&HistoricalAwards=false
• 关键词: Algebra; Blueprinted; Geometry; Combinatorics; Matroids

Lines and planes in a 3-dimensional space are examples of "linear subspaces", a fundamental notion in mathematics. Matroids are discrete analogs of linear subspaces, and they play an increasingly important role in modern mathematics. The PI and Nathan Bowler recently put forth a new point of view in which matroids and linear subspaces are both special cases of a more general class of objects, thus making the analogy between matroids and linear subspaces into something much more precise. Moreover, the PI and Oliver Lorscheid have demonstrated that this more general viewpoint has interesting applications to algebra, algebraic geometry, and combinatorics. This project concerns further development, extensions, and applications of the Baker-Bowler theory. The broader impacts of the proposed work will include supervising postdocs, graduate students, undergraduates, and high school students on research projects, and disseminating mathematics through high-quality written exposition and the organization of conferences and workshops.In more detail, the PI and his collaborators will prove new results about matroid representations, including new results concerning representations of 3-connected matroids and quaternary orientable matroids. They will also extend various aspects of the Baker-Bowler theory to orthogonal matroids (also known as even delta-matroids or Coxeter matroids of type D). Finally, they will develop the rudiments of intersection theory for systems of multivariate polynomials over pastures, extending the PI's work with Lorscheid in the univariate case.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.

三维空间中的直线和平面是“线性子空间”的例子，这是数学中的一个基本概念。拟阵是线性子空间的离散类似物，在现代数学中起着越来越重要的作用。PI和Nathan Bowler最近提出了一个新的观点，其中拟阵和线性子空间都是更一般的一类对象的特殊情况，从而使拟阵和线性子空间之间的类比变得更加精确。此外，PI和奥利弗Lorscheid已经证明，这个更普遍的观点有有趣的应用代数，代数几何和组合数学。这个项目涉及贝克-鲍勒理论的进一步发展、扩展和应用。拟议工作的更广泛影响将包括监督博士后，研究生，本科生和高中生的研究项目，并通过高质量的书面博览会和会议和研讨会的组织传播数学。更详细地说，PI和他的合作者将证明关于拟阵表示的新结果，包括关于3-连通拟阵和四元可定向拟阵的表示的新结果。他们还将贝克-鲍勒理论的各个方面扩展到正交拟阵（也称为偶δ-拟阵或D型考克斯特拟阵）。最后，他们将开发牧场上多元多项式系统的交叉理论的雏形，扩展PI与Lorscheid在单变量情况下的工作。该奖项反映了NSF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。

项目成果

On the existence of balanced generalized de Bruijn sequences

关于平衡广义de Bruijn序列的存在性

• DOI: 10.1016/j.disc.2023.113487
• 发表时间: 2023
• 期刊: Discrete Mathematics
• 影响因子: 0.8
• 作者: Baker, Matthew;Mittal, Bhumika;Mouli, Haran;Tang, Eric
• 通讯作者: Tang, Eric

Fusion rules for pastures and tracts

牧场和土地的融合规则

• DOI: 10.1016/j.ejc.2022.103628
• 发表时间: 2023
• 期刊: European Journal of Combinatorics
• 影响因子: 1
• 作者: Baker, Matthew;Zhang, Tianyi
• 通讯作者: Zhang, Tianyi

Balancing Centrifuges with Number Theory

用数论平衡离心机

• DOI: 10.1080/10724117.2022.2092372
• 发表时间: 2022
• 期刊: Math Horizons
• 作者: Baker, Matthew H.