Totally Geodesic Subvarieties in the Moduli Space of Riemann Surfaces

在这项由国家科学基金会资助的研究中，首席研究员试图解决一般动力系统研究中的基本挑战，特别是在多边形中运动的球的轨迹。动力系统是随时间演化的数学对象，在数学应用中无处不在。每当用数学来预测未来时，例如预测天气、股票市场或溶液中粒子的行为，就涉及到动力系统。现实世界中的动力系统，就像刚才提到的，通常是非常复杂的。首席研究员将研究一类以多边形理想台球为模型的简单动力系统，以期理解应用中出现的复杂动力系统。首席研究员将致力于发现和分类台球系统中可能的动态行为范围，并将继续他的研究，探索台球与数论之间的联系。此外，首席研究员将继续他的教育、指导和推广活动，以促进他的工作产生更广泛的影响。该项目旨在解决黎曼曲面及其模空间动力学研究中的基本挑战。这些学科与数学的许多领域都有联系，在映射类的分类、有理映射的动力学和多边形中球运动的轨迹等方面都有重要的应用。特别重要的是模空间的特殊子变体，它们在测地线流下是不变的。这样的亚种很少见，它们的起源也很神秘。首席研究员将进行两项研究。首先，首席研究员和他的合作者将给出特殊亚种的新结构。其次，首席研究员将研究特殊子变量与数论之间的联系，其特定目标是理解这些空间的算术几何。

In this NSF funded research, the principal investigator seeks to address fundamental challenges in the study of dynamical systems in general, and trajectories of a ball moving in polygons in particular. Dynamical systems are mathematical objects which evolve in time, and they are ubiquitous in the applications of mathematics. Whenever mathematics is used to predict the future, e.g. to predict the weather, the stock market, or the behavior of particles in a solution, a dynamical system is involved. The dynamical systems in the real world, like those just mentioned, are often extraordinarily complex. The principal investigator will study a simple class of dynamical systems modeled on ideal billiards in polygons with a view towards understanding the complicated dynamical systems that occur in applications. The principal investigator will work to uncover and categorize the range of dynamical behaviors possible in billiard systems, and will continue his research exploring connections between billiards and the theory of numbers. In addition, the principal investigator will continue his educational, mentoring, and outreach activities to promote the broader impacts of his work. This project seeks to address fundamental challenges in the study of dynamics on Riemann surfaces and their moduli spaces. These subjects have connections with many areas of mathematics and important applications to the classification of mapping classes, the dynamics of rational maps and trajectories of a ball moving in polygons in polygons. Of particular importance are the special subvarieties of moduli space which are invariant under the geodesic flow. Such subvarieties are rare and their origins are mysterious. The principal investigator will pursue two lines of research. First, the principal investigator and his coauthors will give new constructions of special subvarieties. Second, the principal investigator will investigate connections between special subvarieties and number theory, with the particular goal of understanding the arithmetic geometry of these spaces.