Interactions between geometry, topology, number theory, and dynamics

Topology is the study of objects up to stretching, and geometry the study of rigid bodies. Work on this project will lead to advances in our understanding of both subjects by combining surprising relationships between them and deep connections to other areas of mathematics and computer science. Both topology and geometry are playing an increasingly important role in applications such as data mining and engineering design, and this project includes collaboration with computer scientists as well as the development of open-source software for exploring aspects of these problems. Graduate students will be trained in this project. The first topic of this project is effective Mostow rigidity and torsion growth in homology, questions motivated in part by number theory and global analysis. The project will explore how topological and geometric invariants behave under towers of finite covers and other geometric limits, and also study the extent to which different topological, geometrical, and arithmetic notions of complexity coincide for hyperbolic manifolds. The second topic of the project is counting essential surfaces in hyperbolic 3-manifolds and, in particular, divining the basic structures of such counts. This will involve placing these questions into the setting of measured laminations as well as relating them to the work of Mirzakhani and to dynamical questions about orbits of integer points under a family of interval isometries. 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.