Moduli of Differentials

微分的概念可以追溯到微积分的起源。例如，微分的积分可以告诉我们物理世界的距离、面积、体积等。如今，微分学与数学内外的许多领域有着广泛的联系。将具有相似结构的微分收集到一个参数空间(称为模空间)中，并研究该空间的全局性质，进而确定关于每个个体微分的关键信息，这已被证明是非常有用的。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，他计划发现最初不相关的主题之间的隐藏联系，在这些主题中，差异从不同的方面表现出来。拟议中的项目还为学生和博士后研究打开了许多大门。首席调查员将继续把他的研究与本科生、研究生和博士后培训以及会议组织结合起来。特别是，他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。黎曼曲面上的微分模空间可以根据零点和极点的类型分层。通过改变由差异引起的多边形结构，每一层都配备了一个群体作用。关于表面几何的各种问题归结为理解在作用下的微分层和轨道层。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、动力学和拓扑学的混合技术，研究这些地层和轨道，如它们的体积、测地线和边界行为。最终的目标是在模空间上建立微分的动态不变量与交理论之间的对应关系。此外，PI计划使用所获得的结果来分析模空间的代数几何性质，如二元类型、紧致和重言环。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.