Combinatorics of Complex Curves and Surfaces

二维瓦片是连接许多数学和物理领域的支点。瓦片的简单性很容易想象和吸引人，它吸引了各级数学家、艺术家、建筑师和普通公众。这个项目的目标是(1)在代数曲线和曲面的最新数学发展的背景下研究平铺，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、博士后和博士后社区，专注于这一思想圈。一个主要的研究领域将是K3曲面空间的模环紧化。这个项目与V.Alexeev合作，试图建立极化K3曲面普适族的扩展到环形紧化的边界，扩展了以前在2阶和椭圆K3曲面上的工作。该方法采用了球面上积分仿射结构的拼接。第二个主要研究课题是高次微分的模空间。本项目旨在研究差异较大的地层，它们的体积，以及它们与瓷砖计数的关系。与P.Smillie的合作研究了将平面分解成彭罗斯状瓷砖的过程。这种方法是新颖的，需要将赫维茨理论推广到一个复杂的叶空间。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.