Representability of infinite matroids and the structure of linkages in digraphs

Abstract

This thesis is part of an ongoing project, which tries to extend different aspects of finite matroid theory to the infinite case. The thesis consists of two major parts. In Chapter 2, the problem of representability of infinite matroids over a field is addressed, where we study different aspects of thin sums matroids.We give a characterization of the duals of ordinary representable matroids among thin sums matroids. We show that the class of tame thin sums matroids is closed under duality and so taking minors. As we shall see, most of the matroids associated to graphs turn out to be tame and thin sums representable. So we suggest the class of tame matroids, as a suitably large class of matroids in which one can have a reasonable theory of representability which is preserved under duality. In Chapter 3, we look at another class of matroids namely the class of gammoids. These matroids are usually given via their presentations which are digraphs. As graph properties are usually easy to visualise, we are interested in the interaction of properties of gammoids as matroids and their presentations. To give a taste of what we do, our approach is similar to identifying a desired class of graphs via forbidding graphs as topological minors. Roughly speaking, looking closely at a system of linkable sets with an undesired behaviour, we try to find the substructure in its defining digraph which causes this undesired behaviour, and then study the class of gammoids definable by the digraphs that do not contain this substructure.