CAREER: Geometric Structures, Character Varieties, and Higher Teichmuller Theory

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，感谢格里戈里·佩雷尔曼，重新引起了人们对局部齐次空间的兴趣，也就是在每个点上看起来都一样的空间。PI建议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。为了产生更广泛的影响，PI希望为本科生、研究生、博士后和早期职业数学家创造一个更具包容性的环境。她建议：组织一个指导阅读计划，让本科生和研究生导师为独立项目配对；一个女性几何与拓扑学网络，有一个网站，年会晚宴和夏季静修，参与者将开始相互合作；中大西洋数学联盟项目（Mid-Atlantic Math Alliance Program），旨在建立一个地区导师社区，帮助未被充分代表的少数族裔学生取得事业成功；小木屋会议（Log Cabin Conferences）是指在偏远地区聚集一小群研究人员（其中许多是早期的研究人员），在协作的氛围中学习一个新的主题。此外，她计划继续组织多样性系列讲座，一年一度的中学女生索尼娅日，弗吉尼亚大学本科生数学俱乐部，担任AWM学生分会的指导老师，AWM项目和数学联盟项目的导师，组织几何研讨会，弗吉尼亚拓扑会议和年度研究生阅读课程。双曲结构是具有有趣变形空间的几何结构的典型例子。PI希望使用在双曲结构背景下开发的结果和技术来研究其他几何结构。例如，她计划研究反德西特空间中的模拟结构。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想要研究“更高的Teichmueller理论”，这是一个表面群变成高实秩李群的“好”表示，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。与经典的Teichmuller理论不同，一般来说，这些表示是否为几何结构的完整是未知的。PI想要研究这个问题，以及这些表示的极限描述以及这些空间上的不同拓扑，几何拓扑。PI认为，来自微分几何和低维拓扑的结果和技术将激发与动力系统、李论、复分析，甚至代数几何、数论、表示理论和物理学有着深刻联系的新的研究方向。该奖项反映了美国国家科学基金会的法定使命，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。

In his Erlanger program of 1872, Felix Klein defined geometry to be the study of properties of a space which 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, thanks to Grigori Perelman, renewed the interest in locally homogeneous spaces, that is spaces that look the same at each point. The PI proposes to study 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 PI wants to develop a more inclusive environment for undergraduate students, graduate students, postdocs and early career mathematicians. She proposes to organize: a Directed Reading Program which pairs undergraduate students with graduate mentors for independent projects; a Women in Geometry and Topology network with a website, annual dinners at conferences and summer retreats where participants will start mutual collaborations; a Mid-Atlantic Math Alliance Program to build a regional community of mentors who will work with underrepresented minority students to help them succeed in their careers; Log Cabin Conferences gathering a small group of researchers (many early career) in a remote location to learn a new topic in a collaborative atmosphere. In addition, she plans to continue organize the Diversity Lecture Series, an annual Sonia Day for middle school girls, the Math Club for undergraduate students at UVa, to be faculty advisor for the AWM Student Chapter, mentor for the AWM program and for the Math Alliance program, to organize the Geometry Seminar, the Virginia Topology Conference and annual graduate reading courses.Hyperbolic structures are the prototypical example of geometric structures with interesting deformation spaces. The PI wants to use results and techniques developed in the context of hyperbolic structures for studying other geometric structures. For example, she plans to investigate analogue structures in anti-de Sitter space. A lot of 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 wants to study "higher Teichmueller theory," that is "nice'" 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. Differently from classical Teichmuller theory, it is not known, in general, if these representations are holonomies of geometric structures. The PI wants to study this question, together with the description of limits of these representations and a different topology on these spaces, the geometric topology. The PI thinks 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.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.