13 Aug 2014

Can you solve the hardest maths problems ever?

As this year’s Fields Awards prove, maths is notoriously hard to penetrate. These are the problems that have been puzzling mathematicians for centuries.

The Fields Medal for excellence in mathematics is often referred to as the Nobel prize of the mathematics world. But unlike the Nobel prize for physics, which was last year awarded to the men behind CERN’s large hadron collider, even attempting to lift the veil of understanding and come to terms with why the Fields Medal winners are worthy is almost impossible.

The good news is that Mark Ronan, honorary professor of maths at UCL, says that we are not alone. “Even when people explain these things, the explanations are quite technical, so you don’t always grasp what’s happened,” he told Channel 4 News. “Until you’ve actually read an article by them, or heard them speak, you don’t know much about it.”

But Terry Lyon, president of the London Mathematical Society (LMS), says there is merit in not pandering to the wider public. “The Fields Medal is absolutely the most prestigious,” he told Channel 4 News. “They both (Nobel and Fields) have outstandingly careful referring….Here, the maths come first, the news comes second.

Fermat’s famous Last Theorem

Maths is an ancient discipline, around 4,000 years old, and there are some theories that have remained unresolved for hundreds of years.

The longest-standing unresolved problem in the world was Fermat’s Last Theorem, which remained unproven for 365 years. The “conjecture” (or proposal) was established by Pierre de Fermat in 1937, who famously wrote in the margin of his book that he had proof, but just didn’t have the space to put in the detail.

In 1995 Andrew Wiles published his own proof – and famously said he’d need more than a margin to prove his point.

53 + 47 = 100 : simples?

But those itching for their Good Will Hunting moment, the Guinness Book of Records puts Goldbach’s Conjecture as the current longest-standing maths problem, which has been around for 257 years.

It states that every even number is the sum of two prime numbers: for example, 53 + 47 = 100. So far so simple. But due to the infinite nature of the sequence of numbers, it has so far been impossible to prove for definite.

The conjecture has spawned a novel by Apostolos Doxiadis, Uncle Petrov and the Goldbach Conjecture, and has been puzzled over for centuries. But as Professor Ronan says: “Nobody’s going to tell you they’ve been spending years trying to prove it.”

Numbers ‘through a telescope’

But when it comes to the most significant, unproven theorem, most agree that it is the Riemann Hypothesis, which was posed by the German mathematician Berhard Riemann in 1859. It states that the nontrivial roots of the Zeta function are of the form (1/2 + b I). Or as Professor Lyon explains, it involves looking at the points where a certain function takes the value of 0, as if “through a telescope”, and has been checked for many, many numbers.

“These points are always supposed to be on a line in the plane… they all seem to be lined up as far as the eye can see. You think: ‘It must be true.’ Unfortunately, that’s not a proof.”

Moreso than the Goldbach conjecture, this is considered hugely important, because of the range of implications that would follow if it was proven, rather than just being a hypothesis. “People who have nothing to do with their time do this on computers,” Professor Lyon added.

‘I’m not interested in money or fame’

Much has been made of the fact that a woman has been awarded one of this year’s four awards for the first time in the history of the prize. Maryam Mirzakhani works in the field of geometry and the description about her work reads: “Because of its complexities and inhomogeneity, moduli space has often seemed impossible to work on directly. But not to Mirzakhani. She has a strong geometric intuition.”

But the British community of mathematicians has also been delighted about Martin Hairer’s award – only the eighth Briton to win the prize ever. He works in the field known as “stochastic analysis”. So far, so impossible to understand. But Professor Lyon said his the area builds on a field that has had a huge impact on everything from mobile phones to the stock market.

And experts have even speculated that his work could shed some light on another of those so-far unsolvable problems – the Navier Stokes problem, which is one of six remaining unsolved Millennium Prize Problems, which include the Riemann hypothesis.

But then again, not all mathematicians are in it for the glory. The Russian genius Grigori Perelman famously rejected the Fields Medal – and an additional $1m – in 2010 for proving the Poincare conjecture, then one of the world’s most difficult problems.

“I’m not interested in money or fame,” he said at the time. “I don’t want to be on display like an animal in a zoo. I’m not a hero of mathematics. I’m not even that successful; that is why I don’t want to have everybody looking at me.”