Talk 4: Combinatorial representations

A \emph{matroid} is a structure which generalises the notion of linear independence of vectors in a vector space; a \emph{representation} of a matroid is an embedding in a vector space so that matroid independence agrees with linear independence. Recently, in connection with network coding, along with Maximilien Gadouleau and S{\o}ren Riis, I introduced the notion of a \emph{combinatorial representation} of a family of sets. As well as matroid theory, this theory links with the notions of orthogonal Latin squares. For example, we use the fundamental existence theorem of Richard Wilson to prove that the edge set of a graph has a combinatorial representation over all sufficiently large alphabets. I will also discuss how various matroid and entropic properties play out in this new situation. http://caul.cii.fc.ul.pt/slides/PeterCameron_talk4.pdf/