Geometry and Dynamics of Holomorphic Geometric Structures

该项目涉及数学科学的基础研究，重点是利用相关数学领域（如复分析）的新技术来理解一类几何物体。该项目的核心是研究模空间——描述几何或机械系统的所有可能形状或配置的对象。这些空间是数学、物理、工程和计算机科学的许多部分的基础，包括理论和应用领域。本课题的基础研究将为利用复分析方法研究几何结构的模空间带来新的见解。该项目的数学可视化元素将提供引人注目的图像和动画，可用于向广大受众说明数学研究的结果。通过其本科生研究部分，该项目还将有助于培养数学技能的劳动力，并为数学科学研究生教育提供更强大的申请人资源。给定属的标记紧实双曲曲面空间可以用曲面群在李群SL（2，R）中的表示空间的连通分量来标识。近年来，作为双曲几何SL(2，R)-表示的潜在推广，离散群的半单李群的Anosov表示类受到了广泛的关注。虽然它已经被证明是相当丰富的，但这个更高级别的故事仍然不完整。这个项目将侧重于所谓的“高级泰奇穆勒理论”的复杂分析方面的基础工作。专注于复杂分析方面，可以使用额外的方法（如多能理论）和新思想的结合（如算子），并且有望比考虑任意半简单群时取得更大的进展。该项目将把本科生研究指导作为一个重要组成部分。PI还将开发用于研究和说明目的的数学可视化和软件工具。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project involves basic research in the mathematical sciences, focusing on understanding a class of geometric objects using novel techniques from related areas of mathematics (such as complex analysis). At its core, the project studies moduli spaces---objects that describe all of the possible shapes or configurations of a geometric or mechanical system. Such spaces are fundamental to many parts of mathematics, physics, engineering, and computer science, including both theoretical and applied areas. The basic research in this project will bring new insights in the use of complex analysis to study moduli spaces of geometric structures. The project's mathematical visualization elements will provide striking images and animations that can be used to illustrate the results of mathematical research to a broad audience. Through its undergraduate research components, the project will also aid in the development of a mathematically-skilled workforce and a stronger applicant pool for graduate education in the mathematical sciences.The space of marked compact hyperbolic surfaces of a given genus can be identified with a connected component of the space of representations of a surface group into the Lie group SL(2,R). The class of Anosov representations of a discrete group into a semisimple Lie group has received much attention in recent years as a potential generalization of the SL(2,R)-representations arising from hyperbolic geometry. While it has already proved to be quite rich, this higher-rank story remains incomplete. This project will focus on foundational work on the complex-analytic side of this so-called "higher Teichmueller theory". Focusing specifically on the complex-analytic aspects allows the use of additional methods (e.g. pluripotential theory) and the incorporation of new ideas (e.g. opers), and is expected to enable more progress than would be possible when considering arbitrary semisimple groups. The project will incorporate undergraduate research supervision as a significant component. The PI will also develop mathematical visualizations and software tools for research and expository purposes.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.