CAREER: Moduli spaces of surfaces

曲面的模空间在数学和理论物理中占有重要的地位，也是不同领域研究人员的交汇点。它们将曲面上的不同几何形状参数化，因此模空间中的一个点编码了曲面可以采用的形状。经典地说，人们考虑均匀弯曲的形状，从双曲几何、复分析、代数几何和矩阵群中产生了非常不同的观点。最近，奇异平坦几何因其与经典模空间的关系以及与动力系统(随时间演化的系统)理论中的重要实例的联系而变得突出。这个项目将通过共享的技术和类比，在五个相互关联的研究项目中推进模空间的研究。这些研究项目的进展将促进对曲面和高维空间几何的理解，并将开启动力系统的应用。这些主题的性质使它们同时进行的调查具有协同性，并允许它们融入教育活动，包括对研究生的培训和指导，与本科生的垂直整合研究，以及为博士项目开发一门新课程。这项针对本科生的研究计划将包括一项积极主动的招聘战略，旨在提高历来在数学领域代表性不足的群体成员的参与度。本科生将被鼓励参加MathCorp暑期推广计划，较高年级的参与者将接受指导培训。将开展的五个研究计划如下。首先，PI将在Teichmüler空间中建立拟凸余有界平面，并通过将动力学得到的分量粘合在一起，最终建立映射类群的凸余紧曲面子群。其次，PI将通过对特殊结构的分类和研究低亏格模空间来确定是否存在阶至少为3的平移曲面的非平凡轨道闭包。第三，利用Mirzakhani的迹公式和功，PI将显示典型的高亏格表面是良好的光谱扩展器。第四，PI将发展Patterson-Sullivan理论来映射不可定向曲面的类群。第五，PI将制定在扩张表面背景下的独特遍历性标准。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。

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.