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"I think it is true that most people know that there is research in
physics and chemistry -- there are new thing to be discovered -- but
that they view mathematics as a working language that has been worked
out once and for all by the Ancient Greeks, with a few bits by Sir
Isaac Newton, and that what they learnt at school was mathematics.
So how can you go on doing mathematics? People ask me ``What do you do?
How can you discover theorems?'' It is a very difficult concept that it is
still developing. People think of it as a language that has all been
written down.
"I was always very attracted to geometry: I liked geometry because of its
concrete nature, but I realised that you had to apply a variety of tools
in order to make progress. That became the guiding thread through most
of my subsequent life. I'm still doing geometry in the broad sense,
but I discovered that to do geometry you had to do almost everything
else as well: you get involved with topology, differential geometry,
algebra, mathematical physics, you name it: but in some sense I still
regard myself as a geometer.
"Most of my mathematics takes place in my head: I don't write much on
paper, not until it is all finished, and then as you're trying to go
through it all and organise it, you explain it to somebody else, and that
naturally evolves into a discussion, and then you focus on particular
problems, and so on. Mathematics is normally a solitary exercise.
You sit and you think hard for an hour. It gets a bit boring, so a bit
of social interaction adds a whole new dimension and makes life so much
more interesting and attractive. Some people don't collaborate and work
by themselves, but I find interaction one of the satisfying parts."
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