Deformation, topology and geometry in low dimensions

曲面理论与数学的各个部分都有很深的相互作用。二维拓扑、几何、群论、复分析和动力学的融合是丰富结构的源泉，并通过直接和类比的方式影响着许多领域。由于曲面简单而灵活，它们允许形状的大参数空间。对这些空间的研究在所有维度的拓扑学和几何学中都是直接有用的。PI将探索曲面形状空间的几何形状，以及在这个更抽象的空间内的流动与从二维结构族构建三维结构的方式之间的关系。通过类比，二维和三维的想法也启发了相关领域的强大技术，如几何群论和动力学。PI的部分工作将探索其中的一些联系。受资助的研究生除了研究工作外，还将接受数学教育和推广方面的培训，为他们通过高等教育和其他领导角色为社会做出贡献做好准备。PI及其学生和合作者的其他活动将包括针对当地学童及其家庭的公共教育活动，将尖端的数学知识带到更大的纽黑文社区。PI计划研究曲面的Teichmuller空间的Weil-Petersson几何的一些方面，以及它与3-流形理论的联系，以及更广泛地与其他环境中的集体行动的联系。最近PI和合著者的工作在研究Teichmuller空间上的Weil-Petersson度量的测地线流以及它与纤维三维流形的结构之间的关系方面提供了一些新的启发。考虑到这些结果，PI计划更详细地研究TeichMuller空间的粗略结构，因为它的渐近锥体的立方体和度量结构是可见的。这一观点提出了关于Teichmuller空间中极小曲面的有趣问题，并为解决WP测地线流的可见性问题提供了途径。Weil-Petersson几何与纤维三维流形结构的新联系和瑟斯顿关于同调的范数暗示了PI计划要研究的更深层次的结构。在三维流形理论中，曲面分解、曲线复形的结构和映射类群与双曲几何之间的深层相互作用一直是具有挑战性的问题。PI将致力于开发一个更完整的三维流形正则分解理论，重点研究Heegaard分裂、纤颤和次表面投影之间的相互作用。他将研究寻找双曲流形的统一模型和获得蒙皮映射的先验界限的问题。Heegaard分裂和手柄的设置通过双曲空间中的图形折叠技术，连接到一种关于自由群的特征种类的新方法。其他项目(有些与学生一起)包括研究分层双曲空间、绘制无限类型曲面的类群以及双曲地震和Teichmuller周期流之间的关系。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。

The theory of surfaces interacts deeply with all parts of mathematics. The confluence of topology, geometry, group theory, complex analysis and dynamics in dimension two is a constant source of rich structure and influences many fields both directly and by analogy. Because surfaces are simple and flexible, they admit large parameter spaces of shapes. The study of these spaces is useful directly in topology and geometry in all dimensions. The PI will explore the geometry of the space of shapes of a surface, and the relationships between flows within this more abstract space and the ways in which three-dimensional structures are built out of families of two-dimensional structures. By way of analogy, ideas in two and three dimensions have also inspired powerful techniques in allied fields such as geometric group theory and dynamics. Part of the PI's work will explore some of these connections. Graduate students funded by the grant will, in addition to their research work, be trained in mathematics education and outreach, preparing them for contributions to society through higher education and other leadership roles. Additional activities of the PI and his students and collaborators will include public educational activities for local school children and their families, bringing cutting-edge mathematical knowledge to the greater New Haven community.The PI plans to study a number of aspects of the Weil-Petersson geometry of the Teichmuller space of a surface, and its connections to the theory of 3-manifolds and more broadly to group actions in other settings. Recent work of the PI and coauthors has shed some new light on the study of the geodesic flow of the Weil-Petersson metric on Teichmuller space, and its connection to the structure of fibered 3-manifolds. With these results in mind, the PI plans a more detailed study of the coarse structure of the Teichmuller space as made visible by the cubical and metric structure of its asymptotic cone. This point of view suggests interesting questions about minimal surfaces in the Teichmuller space, and avenues toward resolving the visibility problem for the WP geodesic flow. New connections of Weil-Petersson geometry to the structure of fibered 3-manifolds and Thurston's norm on homology hint at deeper structure which the PI plans to investigate. In the theory of 3-manifolds, the deep interaction between decompositions along surfaces, structure of curve complexes and mapping class groups, and hyperbolic geometry continues to provide challenging questions. The PI will work on developing a more complete theory for canonical decompositions of 3-manifolds, focusing on the interactions of Heegaard splittings, fibrations and subsurface projections. He will study the problem of finding uniform models for hyperbolic manifolds and obtaining a priori bounds on skinning maps. The setting of Heegaard splittings and handlebodies connects to a new approach on character varieties of free groups, via a technique of graph folding in hyperbolic space. Additional projects (some with students) include the study of hierarchically hyperbolic spaces, mapping class groups of infinite-type surfaces, and relations between hyperbolic earthquakes and the Teichmuller horocyclic flow.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.