Lecture 1: November 13, 2017. Time and location: TBA
Abstract: Hyperbolic geometry is the richest and most interesting of Thurston’s eight geometries for 3-manifolds. A good understanding of the ways in which hyperbolic geometry interacts with topology in three dimensions also informs our understanding of many related fields, such as geometric group theory and complex dynamics. We will give a brief introduction to this subject, trying to focus on examples, geometric intuition and overall structure.
Lecture 2: November 14, 2017. Time and location: TBA
Abstract: One can probe the geometry of a 3-manifold by mapping in surfaces in different ways: conformal boundaries at infinity give us a parametrization of families of 3-manifolds using classical Teichmuller space, and Thurston’s pleated surfaces relate the varying geometry on the interior to 2-dimensional combinatorial and geometric structures. We will explore these notions and explain a little about how they provide a complete set of invariants for deformation spaces of 3-manifolds.
Lecture 3: November 15, 2017. Time and location: TBA
Abstract: While the theory has had many successes, we are still far from having a complete “effective” recipe for predicting the geometry of a hyperbolic 3-manifold from its topological description. I will describe some ongoing work in this direction, and some remaining questions.
Virginia Mathematics Lectures archive