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

在1872年的Erlanger程序中，Felix Klein将几何定义为研究在其对称群下不变的空间的性质。1935年，查尔斯·埃雷斯曼开始研究几何结构的变形空间，他提出的问题是，在某种几何结构上，哪些“形状”可以被“局部模仿”。1982年，威廉·瑟斯顿的几何化猜想，现在成了一个定理，多亏了格里戈里·佩雷尔曼，重新引起了人们对局部齐次空间的兴趣，局部齐次空间是指在每一点看起来都相同的空间。PI建议研究流形上的结构族，以及当一个人扰动它们时它们是如何变化的，特别是关注几何和动力学方面。作为一个更广泛的影响，PI希望为本科生、研究生、博士后和早期职业数学家发展一个更具包容性的环境。她建议组织：定向阅读计划，将本科生与独立项目的研究生导师配对；几何和拓扑女性网络，包括网站、年度会议晚宴和夏季静修活动，参与者将在那里开始相互合作；中大西洋数学联盟计划，建立一个区域导师社区，这些导师将与未被充分代表的少数族裔学生合作，帮助他们在职业生涯中取得成功；Log Cabin会议，将一小群研究人员(许多是职业生涯早期的人)聚集在偏远的地方，在合作的氛围中学习新的主题。此外，她计划继续组织多样性讲座系列，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学生分会的教师顾问，AWM计划和数学联盟计划的导师，组织几何研讨会，弗吉尼亚拓扑学会议和年度研究生阅读课程。双曲结构是几何结构具有有趣变形空间的典型例子。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。许多形变空间是由流形的基本群表示成Lie群的空间产生的，所以PI也计划继续研究自由群的特征标集的动态分解，以及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”，即曲面群到更高实数阶的李群的“尼斯”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表示被证明是一阶李群的凸余紧子群在高阶李群的离散子群的背景下的推广，PI计划使用为Klein群开发的技术来研究Anosov表示的极限。与经典的泰希穆勒理论不同，一般来说，这些表示是否是几何结构的完整并不为人所知。PI想要研究这个问题，以及这些表示的极限的描述和这些空间上的一种不同的拓扑-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启发与动力系统、李理论、复分析甚至代数几何、数论、表示理论和物理有深刻联系的新的研究方向。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.