In this project you will work on the interface of topological graph theory, matroid theory, and algebraic combinatorics.
Matroids are combinatorial objects that simultaneously generalise vector spaces and graphs. There is a symbiotic relationship between graph theory and matroid theory theory, with results in either area informing results in the other. Recent advances have indicated that such a relationship holds between topological graph theory (i.e., the study of graphs that are embedded in surfaces) and a generalisation of matroids called delta-matroids.
During this project you will work on problems in both topological graph theory and in matroid theory, and may consider their applications to other areas of combinatorics, such as graph polynomials. An emphasis may be on structural results such as excluded minor theorems. Key to the development of such results will be the use of the geometric intuition provided by topological graph theory to uncover the structure of generalisations of matroids such as delta-matroids and multi-matroids.