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Totally Geodesic Subvarieties in the Moduli Space of Riemann Surfaces

黎曼曲面模空间中的全测地线子类型

基本信息

批准号：
1708705
负责人：
Ronen Mukamel
金额：
$ 17.3万
依托单位：
William Marsh Rice University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2019-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1708705&HistoricalAwards=false
关键词：
Totally Geodesic Subvarieties Moduli Space

项目摘要

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.

在这项由美国国家科学基金会资助的研究中，首席研究员试图解决动力学系统研究中的基本挑战，特别是球在多边形中运动的轨迹。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。当数学被用来预测未来时，例如预测天气、股市或溶液中粒子的行为，就涉及到一个动力系统。现实世界中的动力系统，就像刚才提到的那样，往往是非常复杂的。主要研究人员将研究一类以多边形理想台球为模型的简单动力系统，以期了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力学行为范围，并将继续他的研究，探索台球和数论之间的联系。此外，首席调查员将继续他的教育、指导和外联活动，以促进他的工作产生更广泛的影响。这个项目致力于解决黎曼曲面及其模空间上的动力学研究中的基本挑战。这些学科与许多数学领域有关，并在映射类别的分类、有理映射的动力学以及球在多边形中运动的轨迹等方面有重要的应用。特别重要的是模空间的特殊子簇，它们在测地线流下是不变的。这样的亚种很罕见，它们的起源也是神秘的。首席调查员将从事两个方面的研究。首先，主要研究者和他的合著者将给出特殊亚簇的新构造。第二，主要研究人员将调查特殊亚簇和数论之间的联系，目的是了解这些空间的算术几何。