During my undergraduate degree, I worked for some time with Simone Santini (Professor at Universidad Autónoma de Madrid) on theoretical computer science. More precisely, on the relation between representations and computability.
In the first half of 2015 we finished writing a mathematical paper on this topic which we tried to publish. This work arose from an idea I came up with in relation to the halting problem and which I shared with Simone, which evolved into a long research and the resulting article. The text can be found for download here. This article considers how representation of abstract concepts affects the computability of problems, creates a formalism in which to fully understand these notions, and establishes a few interesting related results. Among these results, we consider the theorem proving the impossibility to improve (through a change of representation) the computing power of computationally enumerable representations the most important. We also show that all improvements in computing power through representation can be explained by "technical" improvements (defined precisely). Furthermore, we relate representations and their hierarchies to the Turing hierarchy of degrees of recursive unsolvability and show the joining points, similarities and differences between these two concepts.
We tried to publish this work on a couple of theoretical computer science journals, but were not able to. We acknowledge the work of the reviewers, some of which correctly pointed out some of the aspects on which our limited background on the field made our work unconnected and hard to locate. Currently, we have decided to leave the topic where it is at for the time being.
My end of degree dissertation, also on the same topic, can also be downloaded here.