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Moduli of Differentials

微分模

基本信息

批准号：
2001040
负责人：
Dawei Chen
金额：
$ 15万
依托单位：
Boston College
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-01 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001040&HistoricalAwards=false
关键词：
Moduli Differentials

项目摘要

The concept of differentials dates back to the origin of calculus. For instance, integration of differentials can tell us about distance, area, volume, etc, for the physical world. Nowadays the study of differentials has broad connections to a number of fields in and outside of mathematics. It has been proven extremely useful that one should collect differentials with similar structures into a parameter space (called moduli space) and study the global properties of this space, which can in turn determine crucial information about each individual differential. The principal investigator plans to explore new, fascinating properties of differentials as well as their moduli spaces. Moreover, he plans to discover hidden connections between initially unrelated subjects in which differentials appear from different aspects. The proposed project also opens many gates for student and postdoctoral research. The principal investigator will continue to integrate his research with undergraduate, graduate and postdoctoral training as well as conference organizations. In particular, he plans to advise student research projects, design new courses, and organize a series of workshops, with a focus on increasing diversity and supporting underrepresented groups. The moduli space of differentials on Riemann surfaces can be stratified according to the types of zeros and poles. Each stratum is equipped with a group action by varying the polygonal structure induced by a differential. Various questions about surface geometry reduce to understanding the strata of differentials and orbits under the action. The principal investigator plans to employ his expertise in algebraic geometry, combined with mixing techniques from analysis, dynamics and topology, to study these strata and orbits, such as their volumes, geodesics, and boundary behavior. An ultimate goal is to establish a correspondence between dynamical invariants of differentials and intersection theory on moduli spaces. Moreover, the PI plans to use the obtained results to analyze algebro-geometric properties of moduli spaces, such as birational types, compactifications, and tautological rings.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.

微分的概念可以追溯到微积分的起源。例如，微分积分可以告诉我们物理世界的距离、面积、体积等。如今，微分研究与数学内外的许多领域有着广泛的联系。事实证明，人们应该将具有相似结构的微分收集到参数空间（称为模空间）中并研究该空间的全局属性，这反过来可以确定有关每个单独微分的关键信息，这是非常有用的。首席研究员计划探索微分及其模空间的新的、令人着迷的性质。此外，他计划发现原本不相关的主题之间隐藏的联系，其中从不同方面出现差异。拟议的项目还为学生和博士后研究打开了许多大门。首席研究员将继续将他的研究与本科生、研究生和博士后培训以及会议组织相结合。他特别计划为学生研究项目提供建议、设计新课程并组织一系列研讨会，重点是增加多样性和支持代表性不足的群体。黎曼曲面上微分的模空间可以根据零点和极点的类型进行分层。每个层都通过改变差异引起的多边形结构来配备集体行动。关于表面几何的各种问题都归结为理解作用下的微分层和轨道。首席研究员计划利用他在代数几何方面的专业知识，结合分析、动力学和拓扑的混合技术，来研究这些地层和轨道，例如它们的体积、测地线和边界行为。最终目标是建立微分动态不变量与模空间交集理论之间的对应关系。此外，PI 计划利用所获得的结果来分析模空间的代数几何性质，例如双有理类型、紧化和同义反复环。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。