This project is an opportunity to study fundamental concepts of category theory through
the lens of dessins d’enfants (dessins). These are hands-on mathematical objects with
which I can formulate an understanding of category theory. A dessin is a bipartite graph
embedded on a plane. As well as vertices, edges, and faces, the rotation system of the
graph is also important for this project.
The research entails studying the combinatorial aspects of dessins d’enfants to answer
several research problems concerning:
– rigorously defining the class of dessins d’enfants,
– the definition of morphisms of dessins and their properties,
– describing monomorphisms and epimorphisms of dessins,
– investigating whether the category of dessins is a topos,
– and defining the Euler characteristic of a dessin.
There is room to answer further questions that I develop in the research process. These
may concern conjugate dessins, functors, and Riemann surfaces.
COMPLETING THE PROJECT
Whilst there are many aspects to the above research component, the process will be
sequential and my supervisors have assisted in establishing a timeline (see proposal) for
the completion of the research.
A distinguishing feature of this project is that it has been designed by my supervisors as a
semi-group project, this is discussed in the proposal. We have submitted individual
applications, will carry out our own research, and will each submit a unique report in the
end. However, the three of us students will be building our research from the same set of
initial problems. Thus, we will have the opportunity to enrich or research journey through indepth discussions about are research. The collaborative nature of this process (and the
implications on my findings) will be made clear in the statement of authorship on my
I will spend some time at the beginning of the project, reading textbooks and reference
books as an introduction to the research topic, these include Basic Category Theory by
Thomas Leinster, Graph Theory by Reinhard Diestel, and the Handbook of Categorical
Algebra Vol.1 Basic Category Theory by Francis Borceux.
University of Adelaide
Thomas Dee has recently completed his second year studying a Bachelor of Mathematical Sciences (Advanced) at the University of Adelaide. He intends to major in pure mathematics and pursue a career in research.