Deformation, topology and geometry in low dimensions

曲面理论与数学的各个部分有着深刻的相互作用。拓扑学、几何学、群论、复杂分析和动力学在二维空间的融合是丰富结构的持续源泉，并直接或间接地影响着许多领域。由于表面是简单而灵活的，它们可以容纳较大的形状参数空间。这些空间的研究在所有维度的拓扑和几何中都是直接有用的。PI将探索表面形状空间的几何形状，以及在这个更抽象的空间中流动之间的关系，以及在二维结构家族之外构建三维结构的方式。通过类比，二维和三维的思想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。获资助的研究生除从事研究工作外，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和他的学生和合作者的其他活动将包括为当地学校儿童及其家庭的公共教育活动，为更大的纽黑文社区带来尖端的数学知识。PI计划研究表面的Teichmuller空间的Weil-Petersson几何的许多方面，以及它与3流形理论的联系，更广泛地说，与其他情况下的群体行为的联系。PI和合作者最近的工作为研究Teichmuller空间上Weil-Petersson度规的测地线流及其与纤维3流形结构的联系提供了一些新的思路。考虑到这些结果，PI计划对Teichmuller空间的粗糙结构进行更详细的研究，这可以通过其渐近锥的立方和度量结构来显示。这一观点提出了关于Teichmuller空间中最小曲面的有趣问题，以及解决WP测地线流可见性问题的途径。Weil-Petersson几何与纤维3流形结构的新联系以及Thurston的同调范数暗示了PI计划研究的更深层次的结构。在3流形理论中，曲面分解、曲线复合体结构和映射类群以及双曲几何之间的深度相互作用继续提供具有挑战性的问题。PI将致力于为3流形的正则分解发展一个更完整的理论，重点关注heegard分裂、纤维和地下投影的相互作用。他将研究寻找双曲流形的一致模型和获得蒙皮映射上的先验界的问题。heegard分裂和柄体的设置，通过双曲空间中的图折叠技术，连接了一种研究自由群特征变异的新方法。其他项目（一些与学生一起）包括对分层双曲空间的研究，无限型表面的映射类群，以及双曲地震与Teichmuller环流之间的关系。该奖项反映了美国国家科学基金会的法定使命，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。

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.