Dr Johannes CarmesinSchool of Mathematics University of Birmingham Edgbaston, Birmingham B15 2TT, UK 
Room 204, Watson Building EMail: J.Carmesin@bham.ac.uk 
Combinatorics in 3 dimensions
A key goal here is to extend the fundamental methods from Structural
Graph Theory to study 2dimensional simplicial complexes. The starting point for this project is my
3dimensional analogue of Kuratowski's theorem.


Graph Minors and Connectivity
A minor of a graph is obtained by deleting and contracting edges.
The connection of Graph Minor Theory with topology is already apparent from this definition as deletions and contractions are
"dual operations" for plane graphs, and even more so in the
Graph Minor Structure Theorem of Robertson and Seymour . Here I am particularly interested in studying connectivity
and treedecompositions, and using its methods in other areas. New approaches include "local separators" of graphs as well as "angry theorems" for 3connected graphs.


Matroids
A fundamental theorem in Matroid Theory is Whitney's characterisation of graph planarity in terms of matroids.
I extended this theorem to 3dimensional space. Another direction of research is to extend Whitney's theorem to surfaces,
which led to a characterisation of graphs admitting locally planar embeddings.


Infinite Graphs
An important tool to study infinite graphs are ends, which can be seen as boundary points at infinity of the graph.
In my PhD, I proved Halin's endfaithful spanningtree conjecture in amended form.
