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.

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，现在是一个定理，这要归功于格里戈里·佩雷尔曼（Grigori Perelman），使人们对本地均匀的空间产生了兴趣，这在每个点看起来都一样。 PI提议研究结构家庭的流形及其在伴随它们时的变化，尤其关注几何和动态方面。作为更广泛的影响，PI希望为本科生，研究生，博士学位和早期职业数学家开发更具包容性的环境。她建议组织：一个定向的阅读计划，该计划将本科生与独立项目的研究生导师配对；网站上的几何和拓扑网络中的女性，在会议上的年度晚餐和夏季静修会，参与者将开始相互合作；一项大西洋中大西洋数学联盟计划，旨在建立一个区域导师社区，他们将与代表性不足的少数族裔学生合作，以帮助他们在职业生涯中取得成功；小木屋会议在偏远地区聚集了一小群研究人员（许多早期职业），以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，年度中学女生的年度日期，UVA的本科生的数学俱乐部，成为AWM学生分会的教师顾问，AWM学生计划的导师和数学联盟计划的导师，以及组织的几何学研讨会，弗吉尼亚拓扑结构和年度研究生corlorial intrip.变形空间。 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DE保姆空间中的模拟结构。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组的性格品种的动态分解以及具有可压缩边界的双曲线歧管基本群的基本组。最后，PI希望研究“较高的Teichmueller理论”，即表面群体的“不错”表示较高的真实等级和Anosov表示，而Anosov表示，这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。通常，如果这些表示是几何结构的全体性，则与经典的Teichmuller理论不同。 PI希望研究这个问题，并描述这些表示形式的限制以及这些空间（几何拓扑）的不同拓扑。 PI认为，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，数字理论，代表理论和物理学的深入联系的新研究方向。该奖项反映了NSF的法定任务，并通过评估范围来反映出构成群体的支持者，并构成了构成群体的范围。