Bert Gerards
(CWI Amsterdam and Eindhoven University of Technology)
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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. |