Quadratic Equations! (or: what do mathematicians actually do?)

When I tell people that I study mathematics they tend to have one of two reactions:

1. They make some impressed-sounding noise, or mention that they were terrible at maths at school, and then quickly make it clear that they wish to change the subject.

2. They are genuinely interested, and want me to tell them exactly what it is that I study.

The second reaction is the one that I fear most!  And at this point it is usually me who tries to change the subject.  I have an ongoing competition with myself to increase the length of time which I can spend explaining my research to someone before their eyes glaze over and their body language starts to say “I want to be somewhere else now”.  I am currently up to about 15 seconds.  And the subject I am currently working on (somewhere between graph theory and galois theory) is fairly accessible compared to some of the more exotic branches of mathematics!

The main problem in explaining pure mathematics to a non-mathematician is the level of abstraction involved in the subject.  Most people’s view of mathematics is that it deals with numbers, and it is hard for people to imagine what exactly it is that mathematicians do…add and subtract really big numbers?  Many seem to find it difficult to imagine how it could be that all the mathematics that could be done hasn’t been done already.^* People rarely encounter abstract mathematics before university; and for good reason, as the transition from dealing with concrete quantities in a familiar setting, to treating those quantities and that setting as merely one very special case in a vast world of abstraction, can be rather bewildering.

You can’t comb a coconut (and other important facts)

Apologies to my hordes of devoted followers for my recent absence…I’ll make up for it by including some (shamelessly copied and pasted) pictures today.

I’m going to talk a bit about topology, which is a kind of generalisation of geometry; if we imagine every object to be made of some kind of incredibly pliable substance which enables it to be morphed into any other vaguely similar object, then we can study general properties of space rather than being constricted by such trifling matters as shape and form.  The basic rule is: if you can stretch, squeeze or twist (no cutting or glueing allowed!) a shape into another one, then they have the same topological properties.  The classic – and rather over-used – saying is that a topologist can’t tell his coffee cup from his doughnut (they both have one hole).

Topologists spend a lot of their time considering holes, and trying to work out things like how many different ways there are to wrap spheres of different dimensions around each other, and which knots are actually unknots (not knots!).  It is a very visually appealing branch of mathematics (see the link on the right to “sketches of topology” for some nice images) and there are some wonderfully odd constructions with special properties that topologists like to study.