Compactifications of moduli spaces and their tropicalizations

Algebraic geometry is the study of spaces that can be described as solution sets of systems of polynomial equations. Circles, parabolae, and hyperbolae are all examples, from classical plane geometry, of shapes which are governed by defining polynomials, in these cases polynomials in two variables. In algebraic geometry, the deep connection between algebra (the defining polynomials) and geometry (the resulting shape) is key. Both recently and over the past century, a large effort has been focused on the study of some very particular spaces of long-standing interest, called moduli spaces, of curves and abelian varieties. These are parameter spaces for certain kinds of geometric objects, and they have deep connections throughout geometry, as well as to mathematical physics and combinatorics. This project will develop and employ modern techniques to make new progress on the study of such spaces. The project will also provide research training opportunities for students. This research program is centered on compactifications of moduli spaces and their tropicalizations, i.e., the instantiations of these moduli spaces in the field of tropical geometry. The project uses tropical moduli spaces, which are certain polyhedral complexes, as a geometric instantiation of the boundary combinatorics of an appropriately compactified moduli space. The existence of the tropical space allows the application of combinatorial-geometric techniques, as well as connections to the study of the cohomology of arithmetic groups. 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.