Geometry and dynamics on deformation spaces of geometric structures

几何结构变形空间的几何与动力学

• 批准号: 1506920
• 负责人: Sara Maloni
• 金额: $ 14.65万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2016-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1506920&HistoricalAwards=false
• 关键词: Geometry; dynamics; deformation; spaces; geometric

In his Erlanger program of 1872, Felix Klein defined geometry to be the study of the properties of a space that are invariant under its group of symmetries. It was Charles Ehresmann in 1935 who started the study of deformation spaces of geometric structures, asking which "shapes" can be "locally modeled" on a certain geometry. In 1982 William Thurston's Geometrization Conjecture, now a theorem, renewed interest in locally homogeneous spaces, that is, spaces that look the same at each point. This research project studies families of structures on manifolds and how they change when one perturbs them, focusing in particular on geometric and dynamical aspects. As a broader impact, the investigator will involve graduate students in her work, organize activities aimed at junior mathematicians, and deliver lectures outside the University. She also wants to promote the collaborative side of the work, and to create a supportive and attentive environment for members of groups underrepresented in mathematics.The investigator will employ results and techniques developed in the context of hyperbolic structures to study other geometric structures. For example, she will investigate compact or hyperideal convex polyhedra in anti-de Sitter space, a Lorentzian analogue of the hyperbolic space, and the end(s) of complex hyperbolic manifolds. Many deformation spaces arise from spaces of representations of the fundamental group of a manifold into a Lie group, so the PI is also planning to continue the study of the dynamical decomposition of character varieties of free groups, and of fundamental groups of hyperbolic manifolds with compressible boundary. Finally, the PI will study "higher Teichmueller theory", that is, representations of a surface group into Lie groups of higher real rank, and Anosov representations, which are a dynamical analogue of locally homogeneous geometric structures. Since Anosov representations turn out to be generalizations of convex cocompact subgroups of rank one Lie groups to the context of discrete subgroups of Lie groups of higher rank, the PI plans to use techniques developed for Kleinian groups in order to study limits of Anosov representations. It is anticipated that results and techniques coming from differential geometry and low-dimensional topology will inspire new research directions with deep connections with dynamical systems, Lie theory, complex analysis, and even algebraic geometry, number theory, representation theory, and physics.

在1872年的厄朗格计划中，费利克斯·克莱因将几何学定义为研究空间在其对称群下不变的性质。Charles Ehresmann于1935年开始研究几何结构的变形空间，他提出了一个问题，即哪些“形状”可以在某种几何结构上“局部建模”。1982年威廉·瑟斯顿的几何化猜想，现在是一个定理，重新引起了人们对局部齐次空间的兴趣，也就是说，在每个点上看起来都一样的空间。本研究项目研究流形上的结构族，以及当扰动它们时它们是如何变化的，特别关注几何和动力学方面。作为更广泛的影响，研究者将让研究生参与她的工作，组织针对初级数学家的活动，并在大学外发表演讲。她还希望促进这项工作的协作性，并为数学领域代表性不足的群体成员创造一个支持和关注的环境。研究者将利用在双曲结构背景下发展的结果和技术来研究其他几何结构。例如，她将研究反德西特空间中的紧致或超凸多面体，双曲空间的洛伦兹模拟，以及复杂双曲流形的末端。许多变形空间是由流形的基本群表示为李群的空间产生的，因此PI还计划继续研究自由群的特征变分和具有可压缩边界的双曲流形的基本群的动态分解。最后，PI将研究“更高的Teichmueller理论”，即将曲面群表示为更高实秩的李群，以及Anosov表示，这是局部齐次几何结构的动态模拟。由于Anosov表示是将1阶李群的凸紧子群推广到高阶李群的离散子群，因此PI计划使用为Kleinian群开发的技术来研究Anosov表示的极限。预计微分几何和低维拓扑的研究成果和技术将与动力系统、李论、复分析乃至代数几何、数论、表示论、物理学等领域产生深刻的联系，从而激发新的研究方向。