Mathematics
A 1000-page proof of the complicated geometric Langlands conjecture has been published. It could provide key insights across maths and physics – if people can understand it, says Alex Wilkins
MATHEMATICIANS have proved a key building block of the Langlands programme, sometimes referred to as a “grand unified theory” of maths due to the deep links it proposes between seemingly distant disciplines within the field.
While the proof is the culmination of decades of work and is being hailed as a dazzling achievement, it is also so obscure and complex that it is “impossible to explain the significance of the result to non-mathematicians”, says Vladimir Drinfeld at the University of Chicago. “To tell the truth, explaining this to mathematicians is also very hard, almost impossible.”
The programme has its origins in a 1967 letter sent by Robert Langlands to fellow mathematician André Weil that proposed the radical idea that two apparently distinct areas of mathematics, number theory and harmonic analysis, were in fact deeply linked. But Langlands couldn’t actually prove this and was unsure whether he was right. “If you are willing to read it as pure speculation, I would appreciate that,” wrote Langlands. “If not – I am sure you have a waste basket handy.”
This mysterious link promised answers to problems that mathematicians were struggling with, says Edward Frenkel at the University of California, Berkeley. “Langlands had an insight that difficult questions in number theory could be formulated as more tractable questions in harmonic analysis,” he says.
In other words, translating a problem from one area of maths to another, via Langlands’s proposed connections, could provide real breakthroughs. Such translation has a long history in maths – for example, Pythagoras’s theorem relating the three sides of a triangle can be proved using geometry – by looking at shapes – or with algebra, by manipulating equations (see diagram, below).
As such, proving Langlands’s proposed connections has been the goal for multiple generations of researchers, leading to countless discoveries, including the maths toolkit used by Andrew Wiles to prove Fermat’s last theorem. It has also inspired mathematicians to seek analogous links to Langlands’s idea that might help. “A lot of people would love to understand the original formulation of the Langlands programme, but it’s hard and we still don’t know how to do it,” says Frenkel.
