CAREER: Totally Geodesic Varieties in Moduli Space: Arithmetic and Classification

在这项职业奖资助的研究中，PI将寻求解决一般动力系统研究中的基本挑战，特别是多边形台球。 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。 每当数学被用来预测未来时，例如预测天气、股票市场或溶液中粒子的行为，都涉及到动力系统。 真实的世界中的动力系统，就像刚才提到的那些，往往是非常复杂的。 PI将研究一类简单的动力系统，以多边形中的理想台球为模型，以期了解应用中出现的复杂动力系统。 PI将致力于揭示和分类台球系统中可能的动力学行为范围，并将继续探索台球与数论之间的联系。 利用这笔赠款分配的资源，PI还将创建一个研究生课程，主题与他的研究有关。 该课程将在莱斯大学举办的研讨会上达到高潮，在此期间，该领域的领导者有机会与班上的学生互动。 最后，PI将利用这笔赠款为Say STEM夏令营创建数学生物学课程，该夏令营旨在吸引STEM领域通常代表性不足的高中生。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，并在映射类的分类、有理映射的动力学和多边形中的台球等方面有重要的应用。特别重要的是模空间的特殊子簇，它们在测地线流下不变。 这些变种是罕见的，它们的起源是神秘的。 PI将在本提案中开展两项研究。首先，PI将研究特殊子簇和数论之间的联系，特别是理解这些空间的算术几何的目标。 第二，PI将寻求发展一个理论，为更高的维特殊subvarieties，以期最终classification.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

In the research funded by this CAREER award, the PI will seek to address fundamental challenges in the study of dynamical systems in general, and billiards in polygons in particular. Dynamical systems are mathematical objects which evolve in time, and they are ubiquitous in the applications of mathematics. Whenever math 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 PI 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 PI 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. Using the resources allotted by this grant, the PI will also create a graduate course on topics relevant to his research. The course will culminate in a workshop hosted at Rice University during which leaders in the field have a chance to interact with students in the class. Finally, the PI will use funding from this grant to create a curriculum in mathematical biology for the Say STEM Camp, a camp aimed at engaging high school students in groups typically underrepresented in STEM fields.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 billiards 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 PI will pursue two lines of research in this proposal. First, the PI will investigate connections between special subvarieties and number theory, with the particular goal of understanding the arithmetic geometry of these spaces. Second, the PI will seek to develop a theory for higher dimensional special subvarieties with a view towards eventual classification.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.