Geometric limits in low dimensional geometry and topology

The central theme of this project is that certain classes of mathematical shapes can be understood better by analyzing other shapes that they approximate. For instance, a sphere with a very large radius, like the earth, approximates a flat plane from the perspective of a person standing upon it. Planar geometry then informs the study of spherical geometry. The particular shapes considered in this project are `hyperbolic manifolds', which have been among the most important objects in theoretical geometry for the past hundred years. To supplement the research component of the project, the PI will continue to develop inquiry-based learning (IBL) courses at Boston College, will revise and disseminate his online book `Geometry in 2 dimensions', and run learning workshops for early career mathematicians. In low dimensional geometry and topology, one often studies a sequence of Riemannian manifolds by passing to an appropriate `geometric limit' manifold. For instance, this technique is essential in Thurston's program to understand 3-dimensional manifolds via hyperbolic geometry, and in the proof of his Geometrization Conjecture by Perelman in 2003. This current project is centered on the structure and applications of geometric limits, especially in hyperbolic geometry. Specifically, the PI will study the global topology of the associated `space of all hyperbolic manifolds', continuing his previous work with Lazarovich-Leitner and Warakkagun, and will use geometric limits and their probabilistic cousins `Benjamini-Schramm limits' to study the relationship between the `rank' of a closed hyperbolic 3-manifold and its geometry, extending his previous work with Souto and Abert et al. 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.