As of the date this is published, I'm still working on putting my thesis together. While I don't have a preprint to share my results with you yet, I wanted to give people an idea of what I've been working on, even if you haven't seen me give a talk.
I gave a talk on my research back in May. It was at the Midwest Dynamical Systems Early Career Conference. I've put my notes below, or you can have a look at a pdf here. Please feel free to email me if you have any questions or would like to chat about math!
The big idea here is that in the setting of negatively curved compact surfaces with Riemannian metrics, the lengths of the shortest closed curve in each homotopy class can give up quite a bit of information about the geometry. In this particular case, a comparison between those lengths on the same surface with two different metrics gives us a comparison between the area under those different metrics. What I've done is show that you don't need the length of every closed geodesic. In fact, you can throw out rather substantial sets as long as they don't grow too quickly! (Like all of the closed geodesics that don't self intersect.)
The theorems I reference by Croke and Dairbekov, and Otal are:
Christopher B. Croke and Nurlan S. Dairbekov, Lengths and volumes in Riemannian manifolds, Duke Mathematical Journal \125 (2004), no. 1, 1-14.
J.P. Otal, Le spectre marque des longueurs des surfaces a courbure negative, Annals of Mathematics 131 (1990), no. 1, 151-160.
I talk about the space of geodesics and the boundary at infinity in two blog posts:
I also talk about geodesic currents in these posts:
More on the specification property in this paper:
Karl Sigmund, On Dynamical Systems with the Specification Property, Transactions of of the American Mathematical Society 190 (1974), 285-289.
Again, if you'd rather look at this as a pdf, it's here!