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

职业：几何结构、特征多样性和高等泰希米勒理论

• 批准号: 1848346
• 负责人: Sara Maloni
• 金额: $ 45万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1848346&HistoricalAwards=false
• 关键词: CAREER; Geometric; Structures; Character; Varieties

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.

菲利克斯·克莱因（Felix Klein）在1872年的埃尔朗格纲领中将几何定义为研究空间在其对称群下不变的性质。1935年，Charles Ehresmann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”。在1982年威廉瑟斯顿的几何化猜想，现在一个定理，感谢格里戈里佩雷尔曼，重新感兴趣的局部齐性空间，这是空间看起来一样，在每一点。PI建议研究流形上的结构族，以及当人们扰动它们时它们如何变化，特别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开发一个更具包容性的环境。她建议组织：定向阅读方案，该方案将本科生与研究生导师配对，开展独立项目;几何和拓扑学妇女网络，该网络有一个网站，在会议上举行年度晚宴，参加者将在夏季务虚会上开始相互合作;大西洋中部数学联盟方案，该方案旨在建立一个区域导师社区，这些导师将与代表性不足的少数民族学生合作，帮助他们在职业生涯中取得成功;小木屋会议聚集了一小群研究人员（许多早期的职业生涯）在一个偏远的地方，在合作的气氛中学习一个新的主题。此外，她计划继续组织多样性讲座系列，每年索尼娅日的中学女生，数学俱乐部的本科生在弗吉尼亚大学，是教师顾问的AWM学生章，导师的AWM程序和数学联盟计划，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的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类似结构。许多变形空间产生于流形的基本群到李群的表示空间，因此PI也计划继续研究自由群的特征簇的动态分解，以及具有可压缩边界的双曲流形的基本群。最后，PI希望研究“更高的Teichmueller理论”，即表面群到更高真实的秩的李群的“好”表示，以及Anosov表示，这是局部均匀几何结构的动力学模拟。由于Anosov表示被证明是一阶李群的凸余紧子群到高阶李群的离散子群的推广，PI计划使用为Kleinian群开发的技术来研究Anosov表示的极限。从经典的Teichmuller理论，它是不知道的，在一般情况下，如果这些表示是holonomies的几何结构。PI希望研究这个问题，以及这些表示的限制和这些空间上的不同拓扑，几何拓扑的描述。PI认为，来自微分几何和低维拓扑学的结果和技术将激发新的研究方向，与动力系统，李理论，复分析，甚至代数几何，数论，表示论，该奖项反映了NSF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进行评估，被认为值得支持的搜索.