CAREER: Moduli Space of Curves and Teichmueller Dynamics

职业：曲线模空间和 Teichmueller 动力学

• 批准号: 1350396
• 负责人: Dawei Chen
• 金额: $ 42.94万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1350396&HistoricalAwards=false
• 关键词: CAREER; Moduli; Space; Curves; Teichmueller

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.

Abelian差异定义了基础黎曼表面上的平坦度量。改变平坦的结构会引起对阿贝尔差异模量空间的作用。这称为Teichmueller Dynamics。关于黎曼表面几何形状的许多问题归结为在这种动态下对轨道的研究。拟议的项目旨在使用代数几何形状中的工具探索Teichmueller动力学，并为Riemann表面模量空间的几何形状开发应用程序。最终目标是在模量空间中建立这些轨道的动力学特性与它们闭合的相交理论之间的对应关系。此外，交点计算可以确定模量空间中轨道闭合的循环类别，从而为理解有效分隔线的锥，曲线锥，食物环结构和模量空间的生育模型提供了关键信息。代数几何形状和动态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，而后者则应用分析工具来描述移动点的时间依赖性。尽管他们最初似乎无关，但主要研究者计划通过构建代数方程来衡量Teichmueller动力学的行为来探索其内部联系。从某种意义上说，这类似于在笛卡尔几何形状中引入坐标。拟议的项目还为学生和博士后研究开放了许多途径。首席研究人员将继续将他的研究与本科，研究生和研究生培训以及研讨会组织相结合。更确切地说，他计划开发学生数学研讨会，为学生研究项目提供建议，在代数几何学上设计新课程，创建一个初级学者访问计划，并组织一系列会议和研讨会，重点关注学生和年轻研究人员。

项目成果

The WYSIWYG compactification

所见即所得的紧凑化

• DOI: 10.1112/jlms.12382
• 发表时间: 2020
• 期刊: Journal of the London Mathematical Society
• 作者: Chen, Dawei;Wright, Alex
• 通讯作者: Wright, Alex