A Field's medal is roughly equivalent to a Nobel prize, except that it's awarded for great work in mathematics. It's not awarded every year, and only mathematicians under 40 are eligible. This makes it very much more difficult for women to win it as its more common for women to take time off full-time work at the time that you need to be working extremely hard on some complicated problem.
I wonder how her first mathematics teacher -- the one who didn't think she had any talent and wasn't surprised when Maryam got bad marks -- wasn't able to see her brilliance.
No-one is really expecting any practical applications of general moduli spaces at the moment, but if anything does come out of this area of mathematics, Maryam Mirzakhani's work has cut through the complexity so clearly that it will be in the centre of whatever happens. It's hard to come up with an equivalent example -- the best I can come up with is comparing it the discovery of fast fourier transform.
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