CAREER: Moduli spaces of surfaces

职业：曲面模空间

• 批准号: 2142712
• 负责人: Alexander Wright
• 金额: $ 50万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2142712&HistoricalAwards=false
• 关键词: CAREER; Moduli; spaces; surfaces

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.

曲面模空间在数学和理论物理学中至关重要，是不同领域研究人员的聚集地，他们对曲面上的不同几何形状进行参数化，以便模空间中的点编码曲面可以呈现的经典形状。 ，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丰富的不同观点，最近，奇异平面几何由于其与经典的关系而受到关注。模空间及其与动力系统理论（随时间演化的系统）中重要实例的联系，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推进对模空间的研究。课程将促进对表面几何和高维空间的理解，并将解锁动力系统的应用。这些主题的性质使它们的同步研究具有协同作用，并允许它们融入教育活动，包括研究生的培训和指导。与本科生垂直整合的研究本科生项目的研究将包括一项积极的招募策略，旨在提高历史上在数学领域代表性不足的群体成员的参与度。夏季推广计划，更高级的参与者将接受指导培训。将进行的五个研究计划如下：首先，PI 将在 Teichmüller 空间中构建拟凸共界平面，并最终构建凸面。其次，PI 将通过对特殊结构进行分类并研究低属模空间来确定是否存在至少 3 阶平移表面的非平凡轨道闭包。第三，PI 将使用 Mirzakhani 的迹公式和工作证明典型的高属曲面是良好的谱扩展器。 第四，PI 将开发用于映射类群的 Patterson-Sullivan 理论。第五，PI 将制定膨胀曲面背景下独特遍历性的标准。​​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。