On March 15, the British mathematician Andrew Wiles was awarded the Abel Prize, the most prestigious among mathematicians, for solving a problem that at one point was considered the world’s hardest. The Abel Prize is awarded by the government of Norway and has been around since 2003. It comes with a purse of 6 million Norwegian kroner (Rs.4.8 crore).
Wiles is a number theorist, currently at the University of Oxford. He became famous in 1994 for developing a general proof of Fermat’s Last Theorem, a deceptively simple problem that had remained unsolved since it was discovered in a book penned by Pierre de Fermat, a French lawyer, in 1667. The problem states that, given an + bn = cn, then a, b and c can’t be positive integers if n > 2. As numerous mathematicians, even of formidable prowess, over the years discovered, the equation’s elegance belied the stupendous reasoning skills required to prove it.
The list of those who took a shot contains some impressive names, including Leonhard Euler, Sophie Germain, Adrien-Marie Legendre, Augustin-Louis Cauchy and Ernst Kummer, among scores of others. And not all their efforts were in vain either. With each failure to breach Fermat’s fortress, they had also invented tools of mathematical investigation that proved very useful in other areas of study, as well as insights that signalled to future gladiators the mathematical avenues to stay away from.
It often happens with the Nobel Prize’s winners, and at times with the Abel Prize’s, that people’s recognition of the scientists as laureates is greater than the recognition of their work. The winner of this year’s Abel Prize, however, easily sidesteps that problem: the history and mathematical complexity of Fermat’s Last Theorem together ensure Wiles will be remembered as a solver of the problem more than the winner of the award, or multiple awards.
He took almost a decade to prove the theorem, and drew on centuries of research to reach his conclusion. In particular, he had to solve another extraordinarily difficult problem in algebraic geometry in order to get to the last theorem, going the last mile with help from a student of his (Richard Taylor, who won the $3-million Breakthrough Prize in Mathematics in 2014). Wiles’s ordeals in going all the way are best understood in terms of a central theme in mathematics called completeness.
Beyond the number line, which we can readily visualise, the study of numbers is marked by multiple abstract concepts. Their resolutions required the invocation of new kinds of numbers. And they were not created willy-nilly, in spurts of creative imagination, but to satisfactorily explain existing problems while remaining self-consistent within the universe of all other numbers. And including such new numbers in this universe makes mathematics more complete.
Needless to say, the satisfaction of completeness is a very arduous process that requires consensus and rigorous verification before it becomes mathematical knowledge. A recent example of this – again motivated by a problem as hard as Fermat’s Last Theorem – involves the Japanese mathematician Shinichi Mochizuki. On August 30, 2012, he uploaded over 500 pages of a ‘solution’, split between multiple papers, to his website, through which he’d claimed to have solved the Oesterlé-Masser conjecture.
Also known as the abc conjecture because of the three principal variables involved in describing the problem, it doesn’t have an easy statement like Fermat’s Last Theorem did. As Davide Castelvecchi articulated it in Nature,
The abc conjecture refers to numerical expressions of the type a + b = c. The statement, which comes in several slightly different versions, concerns the prime numbers that divide each of the quantities a, b and c. Every whole number, or integer, can be expressed in an essentially unique way as a product of prime numbers — those that cannot be further factored out into smaller whole numbers: for example, 15 = 3 × 5 or 84 = 2 × 2 × 3 × 7. In principle, the prime factors of a and b have no connection to those of their sum, c. But the abc conjecture links them together. It presumes, roughly, that if a lot of small primes divide a and b then only a few, large ones divide c.
The problem was first put forth in 1985 by Joseph Oesterlé and then expanded by others, especially David Masser, and so its name. Mochizuki himself has been very reclusive, refusing to give interviews while also maintaining a glassy attitude, claiming that the proof is “complete” and that other proofs wouldn’t be very dissimilar to his. However, mathematicians who are trying confirm the validity of Mochizuki’s methods are stumped by the complexity. As Castelvecchi writes, “To complete the proof, Mochizuki had invented a new branch of his discipline, one that is astonishingly abstract even by the standards of pure maths. ‘Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space,’ number theorist Jordan Ellenberg … wrote on his blog a few days after the paper appeared.” The invented discipline is being called inter-universal geometry.
Whether the claimed proof will eventually be accepted is a matter of determining if Mochizuki’s inventions are required to necessitate completeness. But this isn’t proving easy, as Ellenberg attested. Three gatherings of mathematicians were planned to discuss the math – two have concluded while the third is slated for July 2016, at the Research Institute for Mathematical Sciences, Kyoto. Mochizuki himself has estimated that a verification wouldn’t emerge until later this decade – but if and when it does, it would be comparable in significance to Wiles’s achievement.