LMS Aitken Lectures

Tuesday 18 October, 2011

Mathematical Institute, University of Oxford

Professor Geoff Whittle (Victoria University of
Wellington) will give two talks:

230pm in L3: Well-quasi-ordering Binary Matroids

The Graph Minors Project of Robertson and Seymour is one of the highlights of
twentieth-century mathematics. In a long series of mostly difficult papers they
prove theorems that give profound insight into the qualitative structure of
members of proper minor-closed classes of graphs. This insight enables them to
prove some remarkable banner theorems, one of which is that in any infinite set
of graphs there is one that is a minor of the other; in other words, graphs are
well-quasi-ordered under the minor order.

A canonical way to obtain a matroid is from a set of
columns of a matrix over a field. If each column has at most two nonzero
entries there is an obvious graph associated with the matroid;
thus it is not hard to see that matroids generalise graphs. Robertson and Seymour always believed
that their results were special cases of more general theorems for matroids obtained from matrices over finite
fields. For over a decade, Jim Geelen,
Bert Gerards and I have been working towards
achieving this generalisation. In this talk I will
discuss our success in achieving the generalisation
for binary matroids, that is, for matroids
that can be obtained from matrices over the 2-element field.

In this talk I will give a very general overview of my work with
Geelen and Gerards. I will not
assume familiarity with matroids nor will I assume
familiarity with the results of the Graph Minors Project.

4pm
in L1: Matroid Representation over Infinite Fields

A canonical way to obtain a matroid
is from a finite set of vectors in a vector space over a field F. A
matroid that can be obtained in such a way is said to be
representable over F. It is clear that when Whitney first defined
matroids he had matroids
representable over the reals as his standard model, but for a variety of reasons
most attention has focussed on matroids
representable over finite fields.

There is increasing evidence that the class of matroids
representable over a fixed finite field is well behaved with strong general
theorems holding. Essentially none of these theorems hold if F is
infinite. Indeed matroids
representable over the real-- the natural matroids
for our geometric intuition -- turn out to be a mysterious class indeed. In the
talk I will discuss this striking contrast in behaviour.