Character Varieties and Locally Homogeneous Geometric Structures

特征多样性和局部均匀的几何结构

• 批准号: 1709877
• 负责人: David Dumas
• 金额: $ 25万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-06-01 至 2022-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1709877&HistoricalAwards=false
• 关键词: Character; Varieties; Locally; Homogeneous; Geometric

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.

研究几何或机械物体的所有可能形状是科学和工程中许多问题的基本组成部分，从理解蛋白质折叠或星系形成到编程自动驾驶车辆以导航复杂地形。 这个研究项目将拓宽我们对一类“形状空间”问题的理解，其中几何对象是表面（即平面或弯曲的二维形状）或由二维模式中组装的高维件构建的密切相关的空间。这些都是自然的例子来研究，因为频繁出现的曲面几何在各种各样的数学问题和应用。 除了有助于数学知识，该项目的计算和可视化组件将产生数学对象的惊人图像，这些图像以科学家和非科学家都能欣赏的方式展示其错综复杂的结构和复杂性。它是从流形的基本群到李群的同态。 从几何结构空间到群表示空间的映射总是一个局部同胚，但除了一些经典的例子（如常曲率几何），完整对应的全局行为还没有得到很好的理解。 调查员将有助于我们了解这一基本问题的研究情况下，分析方法可以承担。 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理论的工具进行研究。 在其他情况下，例如紧曲面上的真实的投影结构及其推广，几何结构可以理解为基础流形上的偏微分方程系统的解，允许几何问题通过渐近分析，障碍和最大值原理等技术进行攻击。

项目成果

Opers and Non-Abelian Hodge: Numerical Studies

• DOI: 10.1080/10586458.2021.1988006
• 发表时间: 2020-07
• 期刊: Exp. Math.
• 作者: D. Dumas;Andrew Neitzke

Uniformization of compact complex manifolds by Anosov homomorphisms

• DOI: 10.1007/s00039-021-00572-6
• 发表时间: 2021-08
• 期刊: Geometric and Functional Analysis
• 影响因子: 2.2
• 作者: David Dumas;Andrew Sanders

COARSE AND FINE GEOMETRY OF THE THURSTON METRIC

• DOI: 10.1017/fms.2020.3
• 发表时间: 2016-10
• 期刊: Forum of Mathematics, Sigma
• 作者: D. Dumas;Anna Lenzhen;Kasra Rafi;Jing Tao

Asymptotics of Hitchin’s Metric on the Hitchin Section

希钦截面上希钦度规的渐近

• DOI: 10.1007/s00220-018-3216-7
• 发表时间: 2019
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Dumas, David;Neitzke, Andrew
• 通讯作者: Neitzke, Andrew

Geometry Labs United: An Invitation

联合几何实验室：邀请函

• DOI: 10.1090/noti1733
• 发表时间: 2018
• 期刊: Notices of the American Mathematical Society
• 作者: Athreya, Jayadev;Dumas, David;Goldman, William;Grigorian, Sergey;Guzman, Rosemary;Hieronymi, Philipp;Lawton, Sean;Lukyanenko, Anton;Tyson, Jeremy;Wilson, Aaron
• 通讯作者: Wilson, Aaron