Algebraic topology.
http://en.wikipedia.org/wiki/Algebraic_topology
It's a way of reasoning about spaces using symbols. It also feeds into
(and from) category theory, which is concerned with the relationships
between different kinds of mathematical structures (you could
characterise category theory as a theory of mathematical translation).
Topology is weird enough to begin with that algebraic topology
automatically sounds *very* advanced. (It's certainly a very long way
over my head). I guess the interesting thing about it is that it takes
problems that you might think of as rather concrete in nature - the
properties of *figures* - and turns them into a game played with
abstract tokens, a game which could conceivably be played by - or at
least with the aid of - machines, or other entities without any kind
of spatial awareness (ant colonies, say).
I remember in Gillian Cross's The Demon Headmaster, the brainy heroine
helps one of her foster-siblings with the devilishly hard maths
problems set for them by the titular fiend. "You'll need tensor
calculus for that one!" she gasps at one point. I was curious about
tensor calculus for years after that. Turns out it has a major part to
play in the theory of General Relativity.
Dominic
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