Combinatorics of Complex Curves and Surfaces

二维平铺是连接数学和物理学许多领域的支点。易于形象化和吸引人的简单，平铺吸引了各级数学家，艺术家，建筑师和公众。该项目的目标是：（1）在最近关于代数曲线和曲面的数学发展的背景下研究镶嵌，探索它们与代数，几何和表示理论的联系，（2）向广大受众传播数学思想，提高公众对数学的审美和智力欣赏，以及（3）发展一个活跃而多样化的年轻研究人员社区，一个主要的研究领域将是K3曲面空间的模环形紧化。这个项目，与V. Alexeev联合，旨在建立极化K3曲面的通用族到环形紧化边界的扩展，扩展以前在2度和椭圆K3曲面上的工作。该方法采用平铺的积分仿射结构的领域。第二个主要研究课题是高次微分模空间。这个项目的目的是研究地层的高微分，他们的体积，并与计数的平铺连接。与P. Smillie的联合工作探索了将平面分解成彭罗斯式瓷砖的方法。该方法是新颖的，需要一个推广的赫维茨理论，以一个复杂的维leaf spaces.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.