CAREER: Moduli spaces of surfaces

曲面的模空间在数学和理论物理学中具有核心重要性，并且是不同领域研究人员的聚会场所。他们参数化不同的几何形状的表面上，使一个点在模空间编码的形状，表面可以承担。经典上，人们认为均匀弯曲的形状，从双曲几何，复分析，代数几何和矩阵群中产生了丰富的不同观点。最近，奇异平坦几何由于与经典模空间的关系以及与动力系统（随时间演化的系统）理论中的重要例子的联系而获得了突出地位。该项目将通过共享技术和类比将五个相互关联的研究项目联系在一起，推进模空间的研究。这些研究计划的进展将促进对曲面和高维空间几何的理解，并将解锁动力系统的应用。这些主题的性质使他们的同步调查协同，并允许他们融入教育活动，包括研究生的培训和指导，与本科生的垂直整合研究，以及为博士课程的桥梁开发新课程。与本科生计划的研究将包括一个积极的招聘策略，旨在提高历史上在数学代表性不足的群体成员的参与。我们鼓励本科生参与MathCorp的暑期拓展计划，更高级别的参与者将接受指导培训。将开展的五项研究计划如下。首先，PI将在Teichmüller空间中建立拟凸共有界平面，并最终通过将动力学获得的分量粘在一起来建立映射类群的凸共紧曲面子群。其次，PI将通过对特殊结构进行分类和研究低亏格模空间来确定秩至少为3的平移曲面是否存在非平凡轨道闭包。第三，PI将显示典型的高亏格曲面是良好的谱扩展器，使用迹公式和Mirzakhani的工作。第四，PI将发展Patterson-Sullivan理论，用于不可定向曲面的映射类群。第五，PI将制定一个膨胀曲面中唯一遍历性的标准。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标准进行评估，被认为值得支持。

Moduli spaces of surfaces are of central importance in mathematics and theoretical physics and are a meeting ground for researchers working in different fields. They parametrize different geometries on surfaces, so that a point in the moduli space encodes a shape which a surface can assume. Classically, one considers the evenly curved shapes, with richly different points of views arising from hyperbolic geometry, complex analysis, algebraic geometry, and matrix groups. More recently, singular flat geometries have gained prominence because of their relationship to the classical moduli space as well as their connections to important examples in the theory of dynamical systems (systems that evolve over time). This project will advance the study of moduli spaces in five interrelated research programs tied together by shared techniques and analogies. Progress on these research programs will advance the understanding of the geometry of surfaces and higher dimensional spaces and will unlock applications to dynamical systems. The nature of the topics makes their simultaneous investigation synergistic and allows for their integration into educational activities, including the training and mentoring of graduate students, vertically integrated research with undergraduate students, and the development of a new course for the bridge to PhD program. The research with undergraduates program will include a proactive recruiting strategy designed to improve participation of members of groups historically underrepresented in mathematics. Undergraduate participants will be encouraged to participate in the MathCorp summer outreach program, and the more senior participants will receive training on mentoring.The five research programs that will be undertaken are as follows. First, the PI will build quasi-convex co-bounded planes in Teichmüller spaces, and eventually convex cocompact surface subgroups of mapping class groups, by gluing together components obtained via dynamics. Second, the PI will determine if there is a non-trivial orbit closure of translation surfaces of rank at least 3, by classifying special constructions and investigating low genus moduli spaces. Third, the PI will show typical high genus surfaces are good spectral expanders, using the trace formula and work of Mirzakhani. Fourth, the PI will develop Patterson-Sullivan theory for mapping class groups of non-orientable surfaces. Fifth, the PI will develop a criterion for unique ergodicity in the context of dilation surfaces.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.