Combinatorics of Complex Curves and Surfaces

复杂曲线和曲面的组合

• 批准号: 2201221
• 负责人: Philip Engel
• 金额: $ 20.08万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2024-02-29
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2201221&HistoricalAwards=false
• 关键词: Combinatorics; Complex; Curves; 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.

二维平铺位于连接数学和物理许多领域的支点上。瓷砖易于形象化且简单吸引人，让各级数学家、艺术家、建筑师和公众着迷。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、几何和表示论的联系，（2）向广泛的受众传播数学思想并增加公众对数学的审美和智力欣赏，以及（3）发展一个由年轻研究人员、博士后和博士生组成的活跃且多样化的社区，专注于这一思想圈。一个主要研究领域将是空间的模块化环形压缩的K3 表面。该项目与 V. Alexeev 合作，旨在将通用的偏振 K3 表面系列扩展到环形致密化的边界，从而扩展之前在 2 阶和椭圆 K3 表面上的工作。该方法在球体上采用积分仿射结构的平铺。第二个主要研究课题是高阶微分模空间。该项目旨在研究差异较大的地层、其体积以及与平铺计数的联系。与 P. Smillie 的合作探索了将平坦表面分解为彭罗斯式瓷砖的方法。该方法很新颖，需要将 Hurwitz 理论推广到一个复杂维度的叶空间。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。