Character Varieties and Locally Homogeneous Geometric Structures

研究几何或机械物体的所有可能形状，是许多科学和工程问题的基本部分，从理解蛋白质的折叠或星系的形成，到编程自动驾驶车辆来导航复杂的地形。这一研究项目将拓宽我们对这类“形状空间”问题的理解，在这类问题中，几何对象是表面(即平面或曲线的二维形状)或由以二维模式组装的高维碎片构成的密切相关的空间。这些都是需要研究的自然例子，因为曲面几何经常出现在各种数学问题和应用程序中。除了对数学知识的贡献外，这个项目的计算和可视化组件还将产生引人注目的数学对象的图像，这些图像以一种科学家和非科学家都能理解的方式展示其复杂的结构和复杂性。紧致流形上的局部齐次几何结构决定了完整表示，它是从流形的基本群到李群的同态。得到的从几何结构空间到群表示空间的映射总是局部同胚，但除了一些经典例子(如常曲率几何)外，完整对应的整体行为还没有被很好地理解。研究人员将通过研究可以运用分析方法的案例来帮助我们理解这个基本问题。例如，复李群中表面群的Anosov表示族可以通过Kodaira-Spencer形变理论和复解析Teichmueller理论的工具来研究。在其他情况下，例如紧致曲面上的实射影结构及其推广，几何结构可以通过基本流形上的偏微分方程组的解来理解，从而允许通过渐近分析、障碍和最大值原理等技术来攻击几何问题。

Studying all of the possible shapes of a geometric or mechanical object is a fundamental part of many problems in science and engineering, from understanding the folding of proteins or the formation of galaxies to programming autonomous vehicles to navigate complex terrain. This research project will broaden our understanding of a class of such "shape space" problems in which the geometric objects are surfaces (i.e. flat or curved two-dimensional shapes) or closely related spaces built from higher-dimensional pieces assembled in a two-dimensional pattern. These are natural examples to study because of the frequent appearance of surface geometry in a wide variety of mathematical problems and applications. In addition to contributing to mathematical knowledge, the computational and visualization components of this project will produce striking images of mathematical objects that exhibit their intricate structure and complexity in a way that can be appreciated by scientists and non-scientists alike.A locally homogeneous geometric structure on a compact manifold determines a holonomy representation, which is a homomorphism from the fundamental group of the manifold into a Lie group. The resulting map from the space of geometric structures to the space of group representations is always a local homeomorphism, but apart from a few classical examples (such as constant curvature geometries), the global behavior of the holonomy correspondence is not well understood. The investigator will contribute to our understanding of this basic problem by studying cases in which analytic methods can be brought to bear. For example, families of Anosov representations of surface groups in complex Lie groups are amenable to study through Kodaira-Spencer deformation theory, and through the application of tools from complex-analytic Teichmueller theory. In other cases, such as real projective structures on compact surfaces and generalizations thereof, geometric structures can be understood in terms of the solutions of a system of partial differential equations on the underlying manifold, allowing geometric questions to be attacked through techniques such as asymptotic analysis, barriers, and maximum principles.