Diagonal Grobner Geometry of Generalized Determinantal Varieties

这是一个在代数组合学和几何交叉的项目。代数变分是多项式方程组的解集，是代数几何的中心对象。行列式变量是一类重要的代数变量。它们出现在舒伯特微积分中，这是代数几何的一个分支，在组合正性问题的研究中起着核心作用。PI将研究决定变量的泛化以及控制它们的组合对象。该项目将为本科生、研究生和博士后研究人员提供合作机会。主要的焦点将是三个家族的品种：辛矩阵舒伯特品种，交替符号矩阵品种，和颤动座。PI将通过研究相关的组合结构来发展对角线Gröbner几何理论。PI还寻求显式公式来计算交替符号矩阵变体的Castelnuovo-Mumford正则性。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。

This is a project at the crossroads of algebraic combinatorics and geometry. Algebraic varieties are the solution sets to systems of polynomial equations and are the central objects in algebraic geometry. Determinantal varieties are an important class of such algebraic varieties. They appear in Schubert calculus, a branch of algebraic geometry, that plays a central role in the study of combinatorial positivity questions. The PI will study generalizations of determinantal varieties as well as the combinatorial objects that govern them. This project will provide opportunities for collaboration with undergraduate, graduate, and postdoctoral researchers.The primary focus will be on three families of varieties: symplectic matrix Schubert varieties, alternating sign matrix varieties, and quiver loci. The PI will develop the theory of diagonal Gröbner geometry through the study of related combinatorial structures. The PI also seeks explicit formulas to compute the Castelnuovo-Mumford regularity of alternating sign matrix varieties.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.