The focus of my research has been based upon cluster algebras from triangulated orientable surfaces. These cluster structures largely classify mutation finite cluster algebras; an important class of examples in which the data describing how new cluster variables are obtained remains finite. Moreover, they provide a rich class of fundamental examples to aid understanding of the general theory — the majority of questions/conjectures posed of cluster algebras have first been understood in the surface case.
I have been exploring cluster structures on triangulated non-orientable surfaces — a stucture that falls outside the realm of cluster algebras. Nevertheless, many questions posed to cluster algebras make sense for this modified setting, moreover, the answers often prove to be quite similar. For instance, here I show finite type quasi-cluster algebras have spherical exchange graphs.
Recently, a much broader cluster structure was introduced by Lam and Pylyavskyy, the Laurent phenomenon algebra, specifically designed to produce the Laurent phenomenon. In a series of two papers, I showed that the cluster structure of non-orientable surfaces fall into this setting, providing a rich class of geometric examples to help study Laurent phenomenon algebras.
Surface cluster algebra expansion formulae via loop graphs, 2020,
Positivity for quasi-cluster algebras, 2019,