Positive Vector Bundles in Combinatorics

拟阵是组合对象，出现在许多重要的数学领域，因为它们在不同的数学结构（如图、域扩展、超平面排列、匹配和离散优化）中捕获了“独立性”的概念。利用近年来开发的各种技术，PI旨在深化组合学和代数几何之间的相互作用，无论是在拟阵理论还是在拟阵以外的背景下，如cox组合学和代数统计。PI计划让本科生和研究生参与该项目。PI将在拟阵理论的几个项目中使用“拟阵的同义类”的新框架提出。其中许多项目涉及拟阵数值不变量的新性质，这些性质对拟阵理论中一些长期存在的猜想有影响。这些项目也启发了考克斯特组合学和代数统计学的项目。这些项目的完成将揭示诸如多拟阵和三角拟阵等组合对象的新结构性质，并将告知代数统计中最大似然问题的复杂性。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。

Matroids are combinatorial objects that appear in many important areas of mathematics because they capture the notion of “independence” in diverse mathematical constructs, such as graphs, field extensions, hyperplane arrangements, matchings, and discrete optimizations. Using various techniques developed in recent years, the PI aims to deepen the interaction between combinatorics and algebraic geometry, both in matroid theory and in contexts beyond matroids such as Coxeter combinatorics and algebraic statistics. The PI plans to involve undergraduate and graduate students in the project.The PI will work on several projects in matroid theory are proposed using the new framework of "tautological classes of matroids." Many of these projects concern new properties for numerical invariants of matroids that have implications to some long-standing conjectures in matroid theory. These projects also inspire projects in Coxeter combinatorics and algebraic statistics. Completion of these projects will reveal new structural properties of combinatorial objects such as polymatroids and delta-matroids, and will inform the complexity of maximum-likelihood problems in algebraic statistics.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.