Dynamics and the Classification of Geometries on Manifolds

本项目将对流形上局部齐次几何结构的分类进行研究。这是一个关于几何学和拓扑学的交界点的课题，在研究2维和3维拓扑学中尤为重要。一个简单的例子是，没有公制精确的世界地图集：球体(地球)的拓扑与欧几里得几何不兼容，尽管环面(内管)的拓扑与欧几里得几何兼容。该项目与其他数学领域密切相关，包括群论、代数几何和动力系统。这项研究有一个实验部分，涉及可视化技术。该项目将包括针对这些问题的软件开发，研究生将接受参与研究的培训。研究包括四个主线：(1)具有可解完整的仿射和射影结构的分类；(2)某些几何与辛/接触几何之间的关系；(3)映射类群作用于形变空间和特征变量所产生的动力系统；(4)拓扑自同构的显式构造和更精细的动力不变量的计算。后者涉及变形空间上的自然泊松/辛几何。该项目还包括对学生的指导和培训，以及通过PI所在学院的实验几何实验室进行的外展活动。该奖项反映了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。

This project will conduct research into the classification of locally homogeneous geometric structures on manifolds. This is a subject on the interface of geometry and topology and has been particularly important in the study of topology in 2 and 3 dimensions. A simple example is that there is no metrically accurate world atlas: the topology of the sphere (the earth) is not compatible with Euclidean geometry, although the topology of a torus (an inner tube) is. The project is closely related to other fields of mathematics, including group theory, algebraic geometry, and dynamical systems. The research has an experimental component and involves visualization techniques. The project will include software development for these questions and graduate students will receive training through involvement in the research.The research includes four main threads: (1) the classification of affine and projective structures with solvable holonomy; (2) relations between the certain geometries and symplectic/contact geometry; (3) dynamical systems arising from mapping class group actions on deformation spaces and character varieties; and (4) explicit constructions of topological automorphisms and the computation of finer dynamical invariants. The latter involves natural Poisson/symplectic geometry on the deformation spaces. The project also includes mentoring and training of students as well as outreach activities via the Experimental Geometry Lab at the PI's home institution.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.