CAREER: Algebraic Curves and Their Moduli: Degenerations and Combinatorics

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Algebraic geometry is a central subject in mathematics and has connections and applications to many areas in mathematics, physics, and engineering. Algebraic geometers study spaces called algebraic varieties that are defined by polynomial equations. One powerful method of studying these spaces, as used in this project, is the method of degenerations, where a parametrized family of algebraic varieties breaks into pieces in the limit. Roughly speaking, the idea is that one studies the combinatorics, i.e. the discrete data, of the pieces, in order to deduce things about the more complicated original space. The educational component of the project includes a Women in Algebraic Geometry Workshop and a seminar series on diversity and inclusion in mathematics.The research supported by this award will center on using modern degeneration techniques, especially those from the field of tropical geometry, to study classical spaces from algebraic geometry. In one direction, the PI will use these techniques to study the topology of classical moduli spaces of curves and abelian varieties. In another direction, the PI will also advance our understanding of Brill-Noether varieties using the combinatorics of set-valued tableaux, and investigate questions in tableau combinatorics that were motivated by the program in Brill-Noether theory. This award will also support a Women in Algebraic Geometry workshop and a seminar series on diversity and inclusion in mathematics.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.