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 reals (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.
The slides of the talk are available at http://www.maths.qmul.ac.uk/~pjc/csgnotes/real-rep-uk.pdf