Diagonal Grobner Geometry of Generalized Determinantal Varieties

这是一个处在代数组合学和几何学的十字路口的项目。代数族是多项式方程组的解集，是代数几何的中心对象。行列式簇是此类代数簇的一类重要形式。它们出现在舒伯特演算中，舒伯特演算是代数几何的一个分支，在组合正性问题的研究中发挥着核心作用。PI将学习行列式变元的泛化以及支配它们的组合对象。这个项目将提供与本科生、研究生和博士后研究人员合作的机会。主要关注三类变种：辛矩阵Schubert变种、交替符号矩阵变种和箭图轨迹。PI将通过研究相关的组合结构来发展对角线Gröbner几何的理论。PI还寻求明确的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NSF的法定使命，并通过使用基金会的智力优势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.