Research
Combinatorics on words: My interest on combinatorics on words has been primarily motivated by representing real-world objects within a discrete space. In particular, I am been interested in capturing symmetry on words, such as reflective and, in the multidimensional setting, translational symmetries. Going forward I would like to extend more results from one dimension into the multidimensional setting.
k-centre problem for implicitly defined objects (such as graphs and words): Many classes of combinatorial objects can be represented as a weighted graph using some similarity measure to assign weights to the edges. For large graphs, for instance the set of all words of length n, generating the whole graph is impractical. To this end, we seek to take a set of representative samples from the graph. The idea behind the k-centre problem for implicitly defined graphs is to take k samples from some graphs that allow the local properties to be determined. At present I have worked on this problem for (multidimensional) words, using the overlap distance between subwords as the distance. Going forward I would like to study more complex objects.
Crystal Structure Prediction: During my PhD I have focused on the problem of predicting the structures of Crystals from first principles. My main results have been on the hardness of this problem, and more recently on approaches to solving similarly motivated problems. Move forward I would like to show undecidability for the general version of this problem
Temporal Graphs: I have recently begun working on the problem of harmonious colourings in the setting of temporal graphs. The initial results have shown that this is a highly challenging problem even when the underlying graph is a path. The next steps in the project would be to look at solutions to this problem when each time step has been solved.
Stable Matchings: During my Masters, I worked on the stable matching problem for incomplete lists with ties. My main result was providing new bounds on the number of blocking pairs for maximum matchings in this setting. Moving forward I would be interested in obtaining similar results for more complex settings such as the kidney exchange problem.