Combinatorics of Complex Curves and Surfaces

二维贴图是连接数学和物理许多领域的一个支点。平铺图易于视觉化，简单易懂，吸引了各个层次的数学家、艺术家、建筑师和普通公众。该项目的目标是：(1)在代数曲线和曲面的最新数学发展背景下研究平铺，探索它们与代数、几何和表示理论的联系；(2)向广泛的受众传播数学思想，提高普通公众对数学的审美和智力欣赏；(3)发展一个活跃和多样化的年轻研究人员、博士后和博士生社区，专注于这一思想圈。一个主要的研究领域将是K3曲面空间的模环面紧化。该项目与V. Alexeev合作，旨在将普遍族的极化K3曲面扩展到环面紧化的边界，扩展先前在2度和椭圆K3曲面上的工作。该方法采用球面上的整仿射结构的平铺。第二个主要研究课题是高微分的模空间。该项目旨在研究高差的地层，它们的体积，以及与瓷砖枚举的联系。与P. Smillie的合作作品探索了平面分解成彭罗斯式瓷砖的过程。这种方法是新颖的，需要将Hurwitz理论推广到一个复维叶空间。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.