# Summary

Here's a more in-depth summary of my research interests.

I study **functions on the space of curves on a surface**, with particular interest in** extremal length**.

One aspect is the study of the extension of these functions to the closure of the space of curves, the space of **geodesic** **currents**. In joint work with Dylan Thurston, https://arxiv.org/abs/2004.01550, we extend a broad class of curve functions to the space of geodesic currents. One of the main applications of this is the extension of extremal length to geodesic currents, and counting problems. This is also the main work in my thesis. We are also adding the final touches to a sequel of this paper, where among others, we give a definition of geodesic currents in terms of curve functionals satisfying some axioms.

Currently, I am trying to extend the results in my thesis to open surfaces and other settings, such as actions of hyperbolic groups on **R-trees **(with Misha Kapovich).

With Ivan Telpukhovskiy we are working on obtaining curve counting problems for punctured surfaces via extending truncated length to geodesic currents.

I am also exploring the applications of my result for **Anosov representations**. I am trying to subsume length-like functions coming from Anosov representations under the scope of my extension result.

Related to this, I am interested in **random models **for surfaces and 3-manifolds and** knot complements.** In joint work with Tom Cremaschi, Yannick Krifka and Franco Vargas Pallete (https://arxiv.org/abs/2108.08419), we construct hyperbolic 3-manifold obtained by looking at the complement of the canonical lift in the unit tangent bundle of a random geodesic (in the sense of a family of geodesics converging to the Liouville current). We show that the volume of this hyperbolic 3-manifold has a sublinear lower bound in terms of the length of the random geodesic.

Another aspect is the study is the **systole**, i.e., the shortest curve with respect to a given curve function. In joint project with Franco Vargas Pallete, (https://arxiv.org/abs/1911.09078) we introduce the concept of extremal length systole, and we compare extremal length and hyperbolic length for short pants decompositions. In a joint project (https://arxiv.org/abs/2105.03871), with Maxime Fortier-Bourque and Franco Vargas Pallete we show that the extremal length systole in genus 2 achieves a local maximum at the Bolza surface.

Previously, I've also thought about decorated Bers-Teichmueller theory and the symplectic structure of Bers-Teichmueller space. Some rough notes I wrote on this can be found here.