University College London, 21/09/2018
“The triangulation complexity of 3-manifolds.”
The triangulation complexity of a closed orientable 3-manifold M is the minimal number of tetrahedra in any triangulation of M. It is a natural, but poorly understood, invariant. In my talk, I will explain how it may be computed, to within a bounded factor, for any hyperbolic 3-manifold that fibres over the circle with fibre a given closed orientable surface S. I will show that it is equal, up to a bounded factor, to the translation distance of the action of the monodromy on the mapping class group of S. I will also explain how the methods that we develop can be applied to lens spaces; we determine their triangulation complexity to within a universally bounded factor. All this is joint work with Jessica Purcell.
“Determining the shape of a billiard table from its bounces.”
Consider a billiard table shaped as a Euclidean polygon with labeled sides. A ball moving around on the table determines a bi-infinite “bounce sequence” by recording the labels of the sides it bounces off. We call the set of all possible bounce sequences the “bounce spectrum” of the table. In this talk I will explain why the bounce spectrum essentially determines the shape of the table: with the exception of a very small family (right-angled tables), if two tables have the same bounce spectrum, then they have to be related by a Euclidean similarity. The main ingredient in proving this fact is a technical result about Liouville currents for flat cone metrics. This is joint work with Moon Duchin, Chris Leininger, and Chandrika Sadanand.
“The renormalized volume of quasifuchsian manifolds.”
Quasifuchsian hyperbolic manifolds have infinite volume, but have a
well-defined “renormalized” volume. We will explain a simple, differential-geometric definition of this quantity and explain how it relates to, and brings new insight on, the Weil-Petersson geometry of Teichmüller space.