Translation Surfaces and Their Applications

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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.