Mike Newman (University of Ottawa)
“The missing axiom of matroid theory is (once again) lost forever”

Given a set of vectors in a vector space, what can we say about the collection of subsets that are linearly independent? Choosing different sets of vectors gives different collections of independent subsets, but clearly, we cannot choose a set of vectors so as to obtain any collection of subsets as the independent sets (that's an exercise). Can we describe the "admissible" collections in some way?

Of course, there is the answer we tell our students: namely, the independent subsets are those for which the corresponding system of equations has no solutions over the particular field in question. But we would prefer simple axioms, describing the collection purely as a collection of subsets.

There are simple necessary conditions on the collection of independent subsets (that's an intermediate exercise), that were first explicitly mentioned as such by Whitney in the 1930's; he defined a "matroid" to be a collection of subsets that satisfies these axioms. Clearly then, the collection of linearly independent subsets of a fixed set of vectors is an example of a matroid, but there are (many!) other collections of subsets that do not arise in this way (that's a harder exercise). Can we add some more (simple!) axioms and so characterize exactly those matroids which are, in reality, just the collection of linearly independent subsets of a fixed set of vectors? As Whitney asks: Are real-representable matroids finitely axiomatizable?

It turns out that matroids representable over a finite field are finitely axiomatizable, if the order of the field is less than 5. Rota's conjecture would imply finite axiomatizability with respect to any finite field.

For an infinite field, the situation is more complicated. In the 1970's Vamos said "no" in a paper with an alluring and unambiguous title. However, a careful reading of his paper reveals that his notion of both "matroid" and "axiom" differs significantly from what one would want.

We show in this talk that, under the correct notion of "matroid" and a reasonable definition of "axiom", it is not possible to finitely axiomatize matroids representable over any particular finite field. This talk will not presume knowledge of matroids or axioms, nor completion of the exercises (though I promise that at least one of them is easy).

This is joint work with Geoff Whittle and Dillon Mayhew.