Translation Surfaces and Their Applications

这个项目的主要目标是促进我们对一个基本的数学对象的理解，这个对象被称为“平移面”。“从基本的角度来看，平移表面可以被视为一个多边形，其边缘有一个粘合的配方。 例如，粘合正方形的相对两侧会产生一个环面，这是甜甜圈形状的数学名称。 最令人惊讶的是，这些多边形迅速导致深入的数学研究的前沿，他们交叉了广泛的领域，包括代数，几何和动力系统。 此外，平移曲面已经开始出现在数学物理学中，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和PI已成功地解释给高中学生如何找到一个球的周期轨迹在一个矩形表，使用标准的反射规则，使用平移表面。 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，这是一种在互联网上免费提供的开源软件。 PI已经为本科生开发了关于平移表面的项目。这个项目的目标有三个。 首先，PI将通过分类问题研究平移曲面的轨道闭包的几何形状。 其次，PI将运用他的轨道闭合几何知识来解决模空间和平移表面的动力学问题。 他将通过他提出的一个猜想来解决关于Kontsevich-Zorich cocycle的Lyapunov指数的异常行为的问题。 最后，他将研究有限亏格黎曼曲面上高阶极点的二次微分的结构，这在数学物理中有应用。

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.