Translation Surfaces and Their Applications

平移表面及其应用

• 批准号: 1738381
• 负责人: David Aulicino
• 金额: $ 9.84万
• 依托单位: CUNY Brooklyn College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1738381&HistoricalAwards=false
• 关键词: Translation; Surfaces; Their; Applications

The primary goal of this project is to advance our understanding of a fundamental mathematical object called a "translation surface." From an elementary perspective, translation surfaces can be viewed as a polygon with a recipe for gluing its edges. For example, gluing opposite sides of a square produces a torus, which is the mathematical name for the shape of a doughnut. Most spectacularly, these polygons quickly lead to deep mathematics at the frontiers of research and they intersect a wide range of fields including algebra, geometry, and dynamical systems. Furthermore, translation surfaces have started to appear in mathematical physics with connections to quantum mechanics. From an educational standpoint, polygons arise in high school geometry and the PI has successfully explained to high school students how to find periodic trajectories of a ball on a rectangular table, using standard reflecting rules, using translation surfaces. The elementary nature of polygons allows undergraduates to meaningfully experiment with them using software, such as Sage, which is open source software that is freely available on the internet. The PI has already developed projects for undergraduates concerning translation surfaces.The goals of this project are threefold. First, the PI will study the geometry of orbit closures of translation surfaces through classification problems. Secondly, the PI will apply his knowledge of the geometry of orbit closures to resolve questions concerning dynamics in moduli space and on translation surfaces. He will approach problems concerning the anomalous behavior of Lyapunov exponents of the Kontsevich-Zorich cocycle through a conjecture he proposes. Finally, he will study the structure of quadratic differentials with higher order poles on finite genus Riemann surfaces, which has applications to mathematical physics.

这个项目的主要目标是促进我们对一个叫做“平移曲面”的基本数学对象的理解。从一个基本的角度来看，平移曲面可以看作是一个多边形，它的边缘被粘合在一起。例如，将一个正方形的相对边粘合在一起，就会产生一个环面，这是甜甜圈形状的数学名称。最引人注目的是，这些多边形迅速导致了研究前沿的深度数学，它们交叉了包括代数、几何和动力系统在内的广泛领域。此外，平移曲面已经开始出现在数学物理中，并与量子力学有关。从教育的角度来看，多边形出现在高中几何中，PI成功地向高中生解释了如何使用标准反射规则，使用平移曲面，在矩形桌上找到球的周期轨迹。多边形的基本性质允许本科生使用软件进行有意义的实验，例如Sage，这是一个在互联网上免费提供的开源软件。PI已经为本科生开发了有关翻译表面的项目。这个项目有三个目标。首先，PI将通过分类问题研究平移面轨道闭包的几何形状。其次，PI将运用他在轨道闭包几何方面的知识来解决模空间和平移曲面上的动力学问题。他将通过他提出的一个猜想来探讨有关kontsevic - zorich循环的Lyapunov指数的异常行为的问题。最后，他将研究有限格黎曼曲面上具有高阶极点的二次微分的结构，这在数学物理上有应用。