CAREER: Moduli Space of Curves and Teichmueller Dynamics

阿贝尔微分定义了基础黎曼曲面上的平坦度量。改变平坦结构会导致阿贝尔微分的模空间上的作用;这被称为泰希穆勒动力学。关于黎曼曲面几何的许多问题都归结为对这种动力学下的轨道的研究。拟议的项目旨在探索Teichmueller动力学使用代数几何工具，并开发应用程序的黎曼曲面的模空间的几何。最终的目标是建立这些轨道的动力学性质和它们在模空间中的闭包的相交理论之间的对应关系。此外，相交计算可以确定模空间中轨道闭包的循环类，这反过来又为理解模空间的有效因子锥、曲线锥、Chow环结构和双有理模型提供了重要信息。代数几何与动力系统是现代数学的两个重要分支。前者使用代数（多项式）方程来研究几何结构，而后者应用分析工具来描述移动点的时间依赖性。尽管它们最初似乎无关，但首席研究员计划通过构建代数方程来测量Teichmueller动力学的行为来探索它们的内在联系。在某种意义上，这类似于在笛卡尔几何中引入坐标。拟议的项目还为学生和博士后研究开辟了许多途径。首席研究员将继续将他的研究与本科生，研究生和研究生培训以及研讨会组织相结合。更确切地说，他计划开发一个学生数学研讨会，为学生的研究项目提供建议，设计代数几何的新课程，创建一个初级学者访问计划，并组织一系列以学生和年轻研究人员为重点的会议和研讨会。

An Abelian differential defines a flat metric on the underlying Riemann surface. Varying the flat structure induces an action on moduli spaces of Abelian differentials; this is called Teichmueller dynamics. A number of questions about the geometry of Riemann surfaces boils down to the study of the orbits under such dynamics. The proposed project aims to explore Teichmueller dynamics using tools in algebraic geometry and to develop applications to the geometry of the moduli space of Riemann surfaces. The ultimate goal is to establish a correspondence between dynamical properties of these orbits and the intersection theory of their closures in the moduli space. Moreover, the intersection calculation can determine the cycle class of an orbit closure in the moduli space, which in turn provides crucial information towards understanding the cone of effective divisors, cone of curves, Chow ring structure, and birational models for the moduli space. Algebraic geometry and dynamical systems are two important branches of modern mathematics. The former uses algebraic (polynomial) equations to study geometrical structures, while the latter applies analytical tools to describe the time dependence of a moving point. Despite the fact they initially seem unrelated, the principal investigator plans to explore their inner connections by constructing algebraic equations to measure the behavior of Teichmueller dynamics. In a sense this is analogous to introducing coordinates in Descartes geometry. The proposed project also opens many avenues for student and postdoctoral research. The principal investigator will continue to integrate his research with undergraduate, graduate and post-graduate training as well as workshop organization. More precisely, he plans to develop a student mathematics symposium, advise student research projects, design new courses in algebraic geometry, create a junior scholar visiting program, and organize a series of conferences and workshops with a focus on students and young researchers.