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 年的厄兰格纲领中将几何学定义为对在其对称群下不变的空间属性的研究。 Charles Ehresmann 在 1935 年开始研究几何结构的变形空间，询问哪些“形状”可以在某种几何上“局部建模”。 1982 年，威廉·瑟斯顿 (William Thurston) 的几何化猜想（现在是一个定理），由于格里戈里·佩雷尔曼 (Grigori Perelman) 的贡献，重新激发了人们对局部同质空间的兴趣，即在每个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： 定向阅读计划，将本科生与研究生导师配对进行独立项目；几何和拓扑领域的女性网络，设有网站、年度会议晚宴和夏季静修会，参与者将在其中开始相互合作；中大西洋数学联盟计划，旨在建立一个地区导师社区，与代表性不足的少数族裔学生合作，帮助他们取得职业成功；小木屋会议在偏远地区聚集一小群研究人员（许多是早期职业），在协作氛围中学习新主题。此外，她计划继续组织多样性讲座系列、一年一度的中学生索尼娅日、UVa本科生数学俱乐部、担任AWM学生分会的教职顾问、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。双曲结构是几何结构的典型例子，具有有趣的意义。 变形空间。 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类似结构。许多变形空间是由流形基本群到李群的表示空间产生的，因此PI还计划继续研究自由群特征簇和具有可压缩边界的双曲流形基本群的动态分解。最后，PI 希望研究“更高的 Teichmueller 理论”，即将表面群“很好”地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。由于阿诺索夫表示被证明是一阶李群的凸协紧子群到更高阶李群的离散子群的泛化，因此 PI 计划使用为克莱因群开发的技术来研究阿诺索夫表示的局限性。与经典的泰希米勒理论不同，一般来说，我们不知道这些表示是否是几何结构的完整谱。 PI 希望研究这个问题，以及对这些表示的限制的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成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、表示论和物理学有深入的联系。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。