Combinatorics of Complex Curves and Surfaces

Two-dimensional tilings lie at a fulcrum connecting many areas of mathematics and physics. Easy to visualize and appealing in their simplicity, tilings have fascinated mathematicians at all levels, artists, architects, and the general public. This goals of this project are (1) to study tilings in the context of recent mathematical developments about algebraic curves and surfaces, exploring their connections to algebra, geometry, and representation theory, (2) to disseminate mathematical ideas to a wide audience and increase aesthetic and intellectual appreciation of mathematics in the general public, and (3) to develop an active and diverse community of young researchers, postdocs, and PhD students focusing on this circle of ideas. One primary area of research will be modular toroidal compactifications of spaces of K3 surfaces. This project, joint with V. Alexeev, seeks to build extensions of the universal family of polarized K3 surfaces to the boundary of a toroidal compactification, extending previous work on degree 2 and elliptic K3 surfaces. The approach employs tilings of integral-affine structures on the sphere. The second primary research topic is moduli spaces of higher differentials. This project aims to study strata of higher differentials, their volumes, and the connection with enumeration of tilings. Joint work with P. Smillie explores decompositions of flat surfaces into Penrose-like tiles. The approach is novel, requiring a generalization of Hurwitz theory to one complex-dimensional leaf spaces. 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.