Extremal Combinatorics, Probabilistic Combinatorics

My PhD is at the interface of Graph Minors, Connectivity and Matroids. Recently, the simply connected 2-complexes embeddable in 3-space have been characterised - in a way similar to Kuratowski's characterisation of graph planarity. In other words, there has been found a forbidden minor characterisation of simply connected 2-complexes embeddable in 3-space. In my PhD, I want to investigate this further. This includes applying these techniques, and developing them even further. This work is connected to an ESPRC grant called 'Graph Minors in 3D and connectivity'.