72 The EconomistJuly 22nd 2017
I
MAGINE a frictionless ball rolling
around a billiard table. Next, work out,
on variously shaped tables, which set of
ricochets would merely repeat a pattern,
and which would eventually cover the
whole surface. Full answers are still elu-
sive, but it is the sort of mathematical puz-
zle that outsiders can at least imagine.
By Maryam Mirzakhani’s standards,
such problems were mundane. In her
world, the billiard tables were abstract geo-
metric objects which stretched and
warped. The problems involved notjust
one table but a “moduli space”, ofall possi-
ble such surfaces. Fans called her work on
these mind-spinning abstractions the
“theorem of the decade”.
Until the joy of maths claimed her, she
wanted to be a novelist. Books cost next to
nothing in the Iran of her childhood, and
her earliest ambition was to read every-
thing. Later, her maths had a literary tinge.
She thrilled to the unfolding plot lines in
the problems she studied—though unlike
in literature, she said, they evolved like live
characters. “Just as you start getting to
know them, you look back and realise your
first impression is mistaken.”
By her own account she was a “slow”
mathematician, both in the time it took her
to get started (her first teacher in Tehran
thought she lacked aptitude) and in theway she approached problems: teasing out
solutions by doodling for hours on vast
sheets of paper. These would swathe the
floor of their home, to the delight of her
toddler, and to the amused bewilderment
of her tidy-minded Czech husband. The
point, she said, was not to write down all
the details, but to stay connected with the
problem. She also likened mathematical
inquiry to being lost in a forest, gathering
knowledge to come up with some new
tricks, until you suddenly reach a hilltop
and “see everything clearly”.
But she was quick on other fronts. En-
couraged by her teachers and older broth-
er, she soared through the Iranian educa-
tion system. She was the first girl to
represent the country in the mathematical
Olympiad, winning gold medals in two
successive years. Her beloved abstract sur-
faces can be described geometrically, with
angles, lengths and areas, or algebraically,
with equations. She was fluent in both: a
mathematical polyglot. She found it “re-
freshing” to cross what she dismissed as
the “imaginary” boundaries between dif-
ferent branchesof the subject.
After Harvard and a stint at Princeton,
she ended up at Stanford, winning the
Fields medal—broadly the maths equiva-
lent of a Nobel prize—in 2014, the first
woman to do so since its inception in 1936.Her doctoral thesis alone was an academic
earthquake, leading to papers published in
the three most-admired mathematical
journals. Of her great breakthroughs, per-
haps the mosteasily explained involves
hyperbolic surfaces: roughly, doughnuts
with two or more holes, but where each
point on the surface curves upwards, like a
saddle. These exist, in theory, in infinite va-
rieties. A bigpuzzle involves “geodesic”
lines: the shortest distances between two
surface points. Some may be infinitely
long; others are “closed”, forming loops
with no endpoints. A fascinating and tiny
handful, known as “simple”, never cross
themselves. Her thesis revealed a formula
for how the number of simple closed geo-
desics of a given length rose as that length
increased. Such work might seem abstruse
to outsiders, but uses abound, from cos-
mology to cryptography.
She belied stereotypes. To Americans,
she had to explain that in her native Iran
(unlike Saudi Arabia) women’s education
and careers were notjust tolerated but en-
couraged: her girls’ high school was run by
a national organisation responsible for
hothousing young talent. She was not only
the first woman to win the Fields medal,
but the first Iranian, making her a celebrity
there. Some media flinched piously from
portraying her without a headscarf, a ta-
boo which frayed after her death. Her mar-
riage to a non-Muslim was not recognised,
hampering family visits. Many also be-
moaned her emigration, part of a debilitat-
ing brain drain. She moved to America for
postgraduate study in 1999, a time when to-
day’s anti-Muslim immigration policies
were unimaginable.Drawing a line
She quailed only before the limelight. She
ignored a friend’s e-mail telling her of the
Fields award, assuming it was a practical
joke. In remission from the cancer that
would eventually kill her, she worried that
chemotherapy had left her too weak to at-
tend the awards ceremony.
Men have roughly five in every six
maths-heavy academic jobsin America,
part of a wider puzzle thatneither nature
nor nurture fully explains. One reason
may be that maths talent and female fertil-
ity flower in the same crucial years. She ac-
knowledged the problem of discourage-
ment, but resisted pressure to be a role
model; other women were doing great
things too, and anyway research mattered
more. At conferences, female colleagues,
working in pairs, helped her dodge media
inquiries. While one distracted the jour-
nalist, the other let her ricochet to a more
familiar plane of being. 7Adding up
Maryam Mirzakhani, the world’s leading female mathematician, died on July14th,
aged 40ObituaryMaryam Mirzakhani
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Liu Xiaobo, the subject of the cover story in last week’s
issue, died shortly after it went to press. The Chinese
dissident’s obituary is at economist.com/liu