Deformation, topology and geometry in low dimensions

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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.