Ever since Thurston exhibited the central role played by hyperbolic geometry in the study of three-dimensional manifolds, many links have been established between the topology and geometry of a three-manifold and the set of all representations of its fundamental group into the group of orientation preserving isometries of three-dimensional hyperbolic space.
This project examines these so-called representation varieties for the key class of once-punctured torus bundles. The first aim is to obtain quantitative and qualitative results on the topology and algebraic geometry of these varieties. The second aim is related to the study of profinite completions of the fundamental groups, and examines finite quotients arising from irreducible representations.
The University of Sydney
Youheng Yao is a second-year undergraduate majoring in mathematics and data science at The University of Sydney. He is interested in various fields in pure math, particularly algebra and geometry. He hopes to do the Honours and postgraduate studies in the field in the future.