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Évariste Galois: Died in a duel at the age of 20 but left a lasting legacy

In June the first Breakthrough Prizes in mathematics were announced: \$3 million for each of five recipients, three in America, one in France, and one who divides his time between Britain and America. The new prizes, founded by internet moguls Mark Zuckerberg and Yuri Milner, were announced last December at a ceremony to award similar prizes in physics and the life sciences. Narrowly focused on particular specialisations, these differ from the mathmatics prize, which is spread across the whole subject.

The big difference between mathematics and other sciences is that maths has no need for expensive experimental work, and mathematicians are free to follow their own instincts-trying to propel them in a particular direction would be rather like herding cats. With no need for experimental corroboration and inspiration, mathematics is very much an exercise of the mind, success residing in the proving of theorems.

A theorem might start out as a well-known conjecture — the 1859 Riemann Hypothesis with its implications for the distribution of prime numbers is the most famous — but it can also be far more mysterious, such as an unlooked for connection between two murky areas of mathematics. A great example of this occurred in 1832, when a 20-year-old French mathematician Évariste Galois sat down to write a letter outlining his mathematical results. The next morning he died in a duel. What Galois had done was connect the problem of solving algebraic equations to the study of symmetry. The more symmetries there are among the solutions to an equation, the tighter their bonding and the harder it is to extract them by means of a formula. He gave a precise connection between equations and symmetry, and had this been a known conjecture his work would have received immediate scrutiny. Prizes would have been awarded — though of course he was dead by then — but it was completely unexpected. Reactions only came years later and his fame was entirely posthumous.

Galois's prominence as a political revolutionary reputedly brought 300 people to his funeral, but Zuckerberg and Milner want mathematicians to be well-known for their mathematical prowess. Milner, who quit the world of physics and made a billion dollars in business, says, "Scientists should be better appreciated. They should be modern celebrities, alongside actors and entertainers. We want young people to get more excited."

It's a fine idea, though celebrity status does not suit all mathematicians. Apparently when Andrew Wiles proved Fermat's Last Theorem, a clothing company wanted him to model a pair of jeans — he wasn't interested. And when the Russian mathematician Grigori Perelman won mathematics' highest honour for his proof of the Poincaré Conjecture (about three-dimensional curved spaces) he refused to attend the ceremony or accept the prize money.

Of course the settling of an old conjecture can readily be publicised, but what of the deeper undercurrents that barely cause a ripple on the public surface of mathematics? Unfathomable, even to the educated layman, these are where the new ideas lie. Work on an old conjecture that does not delve into such depths usually results in the efforts being stillborn. A lovely fictional example appears in Uncle Petros and Goldbach's Conjecture. In Apostolos Doxiadis's novel, the protagonist gives up a very promising mathematics career to concentrate all his efforts on the conjecture that every even number is the sum of two prime numbers, but in a William Golding-like descent into madness, he fails.