Beyond fibrations: flows, foliations, and geometry from 3-manifolds to free-by-cyclic groups

Movement of particles can be extraordinarily complicated. The theory of dynamical systems studies such movements by replacing the three-dimensional space with a finite two-dimensional slice. Though tremendously powerful, this technique has its limits. For instance, there is no finite, two-dimensional space that captures all of the original system’s dynamics. This project explores new ways of understanding low-dimensional dynamical systems that are inspired by the classical approach and seeks to unify major threads in the study of structures on three-dimensional spaces. In terms of broader impacts, the PI will continue to co-organize a regional seminar, mentor graduate students, and volunteer mathematics homework help and tutoring during the afterschool program at a local recreation center. The PI’s research divides into three interrelated themes. First, the PI will work on classifying all pseudo-Anosov flows that are transverse to a fixed taut foliation. His approach proceeds by first understanding the universal circles associated to such a foliation and builds on recent advances made in his joint work with Landry and Minsky. Second, the PI will connect the dynamics and topology of these flows and foliations with the hyperbolic geometry of the underlying three-manifold. In particular, the PI and his collaborators recently developed a new method to study endperiodic maps of infinite-type surfaces by producing a ‘pseudo-Anosov–like’ representative associated to the flow. The PI will exploit this new structure by studying the dynamics of the resulting maps and the hyperbolic geometry of their mapping tori. Third, the PI will extend these new techniques into the dynamical study of free-by-cyclic groups and toward the geometric study of mapping class groups. This includes answering long-standing algorithmic questions in mapping class groups and developing a new fixed-point theory of pseudo-Anosov maps, which itself has consequences for hyperbolic geometry. 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.