Diagonal Grobner Geometry of Generalized Determinantal Varieties

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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.