Positive Vector Bundles in Combinatorics

拟阵是出现在许多重要数学领域的组合对象，因为它们在不同的数学构造中捕捉了“独立性”的概念，例如图，域扩展，超平面排列，匹配和离散优化。 使用近年来开发的各种技术，PI旨在加深组合数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之外的背景下，如Coxeter组合数学和代数统计。PI计划让本科生和研究生参与这个项目。PI将致力于拟阵理论中的几个项目，这些项目是使用“拟阵的重言式类”的新框架提出的。“这些项目中的许多涉及拟阵的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理论有影响。 这些项目也激发了Coxeter组合数学和代数统计的项目。 这些项目的完成将揭示新的组合对象的结构特性，如polymatroids和delta-matroids，并将告知代数统计中的最大似然问题的复杂性。该奖项反映了NSF的法定使命，并已被认为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。

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.