This is a lecture about the power of simple ideas in mathematics.

What I like doing is taking something that other people thought was complicated and difficult to understand, and finding a simple idea, so that any fool – and, in this case, you – can understand the complicated thing.

These simple ideas can be astonishingly powerful, and they are also astonishingly difficult to find. Many times it has taken a century or more for someone to have the simple idea; in fact it has often taken 2000 years, because often the Greeks could have had that idea, and they didn't.

People often have the misconception that what someone like Einstein did is complicated. No, the truly earth-shattering ideas are simple ones. But these ideas often have a subtlety of some sort, which stops people from thinking of them. The simple idea involves a question nobody had thought of asking.

Consider, for example, the question of whether the Earth is a sphere or a plane. Did the ancients sit down and think ‘now let's see – which is it, a sphere or a plane?’? No, I think the true situation was that no-one could conceive the idea that the earth was spherical – until someone, noticing that the stars seemed to go down in the West and then twelve hours later come up in the East, had the idea that everything might be going round – which is difficult to reconcile with the accepted idea of a flat earth.