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: 1650811
• 负责人: Sara Maloni
• 金额: $ 11.59万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1650811&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.

Felix Klein在1872年的Erlanger计划中定义了几何形状是对在其对称组组下不变的空间的性质的研究。 1935年，查尔斯·埃斯曼（Charles Ehresmann）开始研究几何结构的变形空间，询问可以在某个几何形状上“局部建模”哪种“形状”。 1982年，威廉·瑟斯顿（William Thurston）的几何化猜想，如今是一个定理，对当地均匀的空间（即每个点看起来相同的空间）提出了兴趣。该研究项目研究了结构家庭的流形以及它们在伴随它们时的变化，特别关注几何和动态方面。作为更广泛的影响，研究人员将让研究生参与她的工作，组织针对初级数学家的活动，并在大学外进行讲座。她还希望促进工作的协作方面，并为数学中代表性不足的群体成员创造一个支持和细心的环境。研究人员将采用双曲线结构中开发的结果和技术来研究其他几何结构。例如，她将在抗DE保姆空间，双曲线空间的Lorentzian类似物以及复杂双曲线歧管的末端研究紧凑或高层凸多面体。许多变形空间来自歧管的基本组的表示空间，因此PI还计划继续研究自由群体角色品种的动态分解，以及具有压缩边界的双曲线歧管的基本组。最后，PI将研究“更高的Teichmueller理论”，即表面组成较高级别的谎言组和Anosov表示，它们是局部同质几何结构的动态类似物。由于Anosov表示是对等级较高等级的谎言组的离散亚组的凸CoCompact子组的概括，因此PI计划使用为Kleinian群体开发的技术以研究Anosov表示的限制。可以预料，来自差异几何形状和低维拓扑的结果和技术将激发与动态系统，谎言理论，复杂分析甚至代数几何形状，数字理论，表示理论和物理学的深入联系的新研究方向。