The Algebra, Blueprinted Geometry, and Combinatorics of Matroids

三维空间中的线和平面是“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，在现代数学中发挥着越来越重要的作用。PI和Nathan Bowler最近提出了一种新的观点，认为拟阵和线性子空间都是一类更一般的对象的特殊情况，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Oliver Lorscheid已经证明，这种更普遍的观点在代数、代数几何和组合学中有有趣的应用。本项目涉及贝克-鲍勒理论的进一步发展、扩展和应用。建议工作的更广泛影响将包括指导博士后、研究生、本科生和高中生的研究项目，并通过高质量的书面报告和组织会议和研讨会传播数学。更详细地说，PI和他的合作者将证明关于阵表示的新结果，包括关于3连通阵和第四纪可定向阵的表示的新结果。他们还将Baker-Bowler理论的各个方面扩展到正交拟阵（也称为D型的偶三角拟阵或Coxeter拟阵）。最后，他们将在牧场上发展多元多项式系统的交集理论的基础，扩展PI与Lorscheid在单变量情况下的工作。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。

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.