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Drinfeld's comment on the Geometric Langlands Proof by Raskin: It's “impossible to explain the significance of the result to non-mathematicians. To tell the truth, explaining this to mathematicians is also very hard, almost impossible.”
From the New Scientist article. https://www.newscientist.com/article/2431964-incredible-maths-proof-is-so-complex-that-almost-no-one-can-explain-it/
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 by dozens of mathematicians 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 from Robert Langlands to fellow mathematician Andre 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.
As such, proving Langlands’s proposed connections has become the goal for multiple generations of researchers and led to countless discoveries, including the mathematical toolkit used by Andrew Wiles to prove the infamous Fermat’s last theorem. It has also inspired mathematicians to look elsewhere for analogous links 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.
One analogy that has yielded progress is reformulating Langlands’s idea into one written in the mathematics of geometry, called the geometric Langlands conjecture. However, even this reformulation has baffled mathematicians for decades and was itself considered fiendishly difficult to prove.
Now, Sam Raskin at Yale University and his colleagues claim to have proved the conjecture in a series of five papers that total more than 1000 pages. “It’s really a tremendous amount of work,” says Frenkel.
#science
From the New Scientist article. https://www.newscientist.com/article/2431964-incredible-maths-proof-is-so-complex-that-almost-no-one-can-explain-it/
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 by dozens of mathematicians 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 from Robert Langlands to fellow mathematician Andre 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.
As such, proving Langlands’s proposed connections has become the goal for multiple generations of researchers and led to countless discoveries, including the mathematical toolkit used by Andrew Wiles to prove the infamous Fermat’s last theorem. It has also inspired mathematicians to look elsewhere for analogous links 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.
One analogy that has yielded progress is reformulating Langlands’s idea into one written in the mathematics of geometry, called the geometric Langlands conjecture. However, even this reformulation has baffled mathematicians for decades and was itself considered fiendishly difficult to prove.
Now, Sam Raskin at Yale University and his colleagues claim to have proved the conjecture in a series of five papers that total more than 1000 pages. “It’s really a tremendous amount of work,” says Frenkel.
#science
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