Geometry and Dynamics of Holomorphic Geometric Structures

该项目涉及数学科学的基础研究，重点是使用数学相关领域（如复分析）的新技术来理解一类几何对象。 在其核心，该项目研究模空间-描述几何或机械系统的所有可能形状或配置的对象。 这样的空间是数学、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。 本计画的基础研究将为复分析在几何结构模空间的应用带来新的见解。 该项目的数学可视化元素将提供引人注目的图像和动画，可用于向广大观众展示数学研究的结果。 通过其本科研究部分，该项目还将帮助发展一个熟练的劳动力和更强大的申请人库的研究生教育在数学科学。空间的标记紧凑双曲曲面的一个给定的亏格可以识别的空间的连通分量的表示的表面群到李群SL（2，R）。离散群到半单李群的Anosov表示类作为双曲几何中SL（2，R）-表示类的潜在推广，近年来受到了广泛的关注。 虽然它已经被证明是相当丰富的，这个更高层次的故事仍然不完整。 这个项目将集中在这个所谓的“更高的Teichmueller理论”的复分析方面的基础工作。专注于复分析方面允许使用额外的方法（例如多能理论）和纳入新的想法（例如opers），并预计将使更多的进展比考虑任意半单群时可能。 该项目将把本科生研究监督作为一个重要组成部分。 PI还将开发数学可视化和软件工具，用于研究和临时目的。该奖项反映了NSF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.