On the Structure of Matrices and Matroids

Bert Gerards (CWI Amsterdam and Eindhoven University of Technology)

Does the class of matrices over a fixed finite field have structure? The class of graphs does; this is the Robertson-Seymour graph structure theorem, which says that for each graph there exists a surface such that all graphs that do not have this graph as a minor embed more or less (in a well specified way) in that surface. Now the notion of minor is geometric rather than graphic and actually the conjecture is that a similar structure theorem holds for matrices over finite fields. It is anticipated that a consequence of such a result is that a minor closed property for binary, ternary, etc. matrices is characterized by a finite list of forbidden minors. Also Rota's conjecture - that representability of a matroid as a matrix over a particular finite field has finitely many forbidden minors - could well follow from such a theory.

This talk is based on joint work with Jim Geelen and Geoff Whittle.

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