RUI: Combinatorial Algebraic Geometry: Curves and Their Moduli

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Algebraic geometry is a central area of mathematics that studies varieties, which are geometric objects defined by systems of polynomial equations. Moduli theory aims to understand specific varieties by considering how they behave in a family of such varieties. In particular, a moduli space consists of all geometric objects of a particular type. This research project consists of problems that arise from the fruitful interactions between algebraic geometry and new developments in other fields of mathematics such as combinatorics, which is concerned with organizing and analyzing discrete structures. This project will fund undergraduate research and the PI will continue efforts towards broadening participation of members of underrepresented groups in the mathematical sciences. The projects aim to better understand moduli spaces that are combinatorially rich in nature and their cohomology theories. Fundamental objects of investigation include degeneracy loci, homogeneous spaces such as Grassmannians and flag varieties, the affine Grassmannian, and the moduli space of curves. More specifically, projects include the study of degeneracy loci and their motivic classes, with applications to Brill-Noether theory; quantum cohomology and quantum K-theory of homogeneous spaces and connections to the affine Grassmannian; base point free classes on the moduli space of curves arising from Gromov-Witten theory and from representation theory; and other problems in algebraic geometry and algebraic combinatorics, including several for undergraduate students.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.