Hyperbolic Manifolds and Their Embedded Submanifolds

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This project will investigate hyperbolic manifolds of finite-volume by understanding the structure of their embedded sub-manifolds. The research combines aspects from multiple fields, including geometric topology, number theory, algebraic geometry, combinatorics, and dynamics. The project will also include participation in activities including mentoring and supporting students and early career mathematicians, participating in public lectures and outreach activities, and organizing events and workshops. This project includes specific research plans for undergraduate and graduate students. The research goals of this project are divided into three main directions. The first project is to continue working in effective virtual properties of 3-manifolds by constructing explicit covers. This project will focus on congruence covers of arithmetic hyperbolic 3-manifolds, where the PI will leverage the rich connection between their geometric and number theoretical properties. The second project is to study codimension-1 embedded sub-manifolds in higher dimensional hyperbolic manifolds. A particular focus of this project is the case of hyperbolic manifolds of dimension 4, to better understand the relationship between the geometry of hyperbolic 4-manifolds and other well-studied 4-manifold invariants. The third project involves the study of embedded surfaces in 3-manifolds and simple closed curves in surfaces through the representations of their fundamental groups. The project also includes broader impact activities aimed at broadening participation among students and junior researchers.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.