Broadly, I am interested in how analysis and geometry interact in the study of complex geometry. You can find some of my work below.

Undergraduate Thesis (TCD):

My undergraduate thesis looked at when one can generalise the Riemann Mapping Theorem of one variable to several complex variables, which classifies domains up to biholomorphism. The key to this is exploiting strong pseudoconvexity of the domain and making use of the Carathéodory and Kobayashi metrics. 

You can find my undergraduate thesis linked here, as well as a poster linked here, and a presentation linked here, which gives a brief overview.

Masters Thesis (Cambridge):

During my master's, I also wrote an essay on the regularity theory of energy-minimising maps between Riemannian manifolds. The Schoen-Uhlenbeck Theorem allows us to get an equivalent regularity theory for harmonic maps as in the case of harmonic functions. However, the curvature of the target domain makes this a much more difficult and delicate question; for instance, there can be certain maps which fail to be smooth at particular points, which we call singularities. As a result, another key focus is classifying the "size" of this singular set, which involves the use of the Hausdorff measure.

You can find my essay linked here.