Mathematician Victor Rotger says that he’ll always remember the primary time he heard concerning the Birch and Swinnerton-Dyer conjecture. She was about to complete her bachelor’s diploma and knocked on a professor’s door to ask if she needed to direct her doctoral thesis. In that workplace, surrounded by towers of papers, the twenty-year-old felt vertigo. “I don’t know what my expression must have been during the minutes I was in Pilar Bayer’s office that first time, but I felt like a parachutist falling over a city I had never been to before,” says the mathematician in a now traditional doc through which he tries to current the conjecture in order that anybody with the will and a minimal of information can perceive it. There are 50 pages of explanations. Another twenty-something mathematician, Raúl Alonso, now faces the identical enigma, one of many seven devilish Millennium Problems, for the answer of which the Clay Institute of the United States presents a reward of 1 million {dollars}.
Alonso, born in Vigo 29 years in the past, doesn’t bear in mind the precise day he heard about Birch and Swinnerton-Dyer’s conjecture, however he does bear in mind the “fascination” he felt. The downside is said to the essence of elliptic curves, mathematical objects already urged by the Greek Diophantus of Alexandria within the third century and which, nearly two millennia later, are elementary within the every day communication of humanity. The WhatsApp utility, for instance, makes use of an elliptic curve, outlined by the equation y² = x³ + 486662x² + x, to calculate a secret key that enables messages to be encrypted and decrypted.
“They can have many shapes,” Alonso explains as he attracts kinds of elliptical curves within the air along with his finger: waves, loops, a mendacity down, a mild hill. He was a toddler prodigy of numbers. When he was 17, he gained a bronze medal on the International Mathematics Olympiad in South Africa, a championship with highschool college students from greater than 100 nations. He later graduated in Mathematics and Physics Engineering from the Polytechnic University of Catalonia and obtained his doctorate from Princeton (USA), the place he was the educational grandson – disciple of a disciple – of the legendary Andrew Wiles, protagonist of one of many best current feats of science.
Wiles locked himself away in secret for seven years, till at some point in 1995 he emerged from his lair and introduced that he had solved the so-called Fermat’s Last Theorem, a riddle scribbled in 1637 by the Frenchman Pierre de Fermat within the margin of a ebook by Diophantus of Alexandria. The theorem, one of the vital well-known in historical past, held that the equality xⁿ + yⁿ = zⁿ is inconceivable if n is an integer higher than 2 and the three letters are optimistic integers. “I have found a really admirable demonstration, but the margin of the book is too small to put it in,” Fermat boasted in a handwritten notice on the sting of a web page. It took humanity greater than three and a half centuries to search out the answer. Alonso, a professor at University College Dublin (Ireland), now faces one other monster of the identical magnitude.
The British Bryan Birch and Peter Swinnerton-Dyer proposed their conjecture in 1965. In her informative talks, the Spanish mathematician Pilar Bayer normally reveals a drawing through which Birch and Swinnerton-Dyer seem round a form of big cake adorned with almonds, absorbed in looking for out what number of nuts they’ve used to brighten it. It is a metaphor for the variety of rational factors on an elliptic curve, the important thing to the conjecture, the entire assertion of which is meaningless to laymen. For his small advances in fixing the issue, Raúl Alonso has simply gained the Vicent Caselles Prize, awarded by the Royal Spanish Mathematical Society and the BBVA Foundation.
Ask. Do you see your self locking your self away for seven years to attempt to resolve the conjecture, like Andrew Wiles did?
Answer. No, the reality is that I do not see myself locking myself up for seven years. I additionally do not know if that’s the regular manner of working immediately. I feel that now it’s extra widespread to collaborate with folks and, maybe, it’s also extra fruitful: to alternate concepts as a substitute of locking your self in and dealing on the issue alone. Each one works as most closely fits him, however I feel that now it’s extra essential to collaborate with different mathematicians. So, no, I’m not going to lock myself up.
P. The Englishman Godfrey Harold Hardy claimed in his ebook Apology of a mathematician that genuine arithmetic is as lovely as poetry and portray and has no functions, no less than imminent. Do you agree?
R. I agree that essentially the most lovely arithmetic is the one that’s investigated based mostly on that magnificence and never based mostly on its potential sensible functions, which doesn’t imply that it’ll not have any. Mathematics might be lovely and have functions on the identical time, I do not suppose it is contradictory.
P. In reality, Hardy revealed his ebook in 1940, within the context of World War II, and was glad that “real mathematics” couldn’t be used to kill your self, however his concept shortly fell aside. [con el desarrollo de la bomba atómica, por ejemplo].
R. Of course.
P. Returning to elliptic curves, maybe essentially the most well-known utility is WhatsApp, which makes use of the one outlined by the equation y² = x³ + 486662x² + x. It is shocking that elliptical curves, which appear so summary, are so essential for secret communications.
R. There are sure issues associated to elliptic curves which can be tough to unravel in an affordable time, and that’s what these cryptographic strategies are based mostly on. [averiguar una clave basada en esa curva podría requerir millones de años con los ordenadores actuales]however I’m not very acquainted with these functions of curves.
P. Could a attainable utility of your work be growing safer communications?
R. There are folks engaged on cryptographic strategies utilizing elliptic curves. You can by no means predict what is going on to occur in years, however as of immediately, the sorts of questions I examine do not appear to have any quick utility in cryptography. Years forward, clearly, nobody can predict what sort of functions it should have.
P. His colleague Francesc Castellà said in an interview with EL PAÍS a decade in the past that the Birch and Swinnerton-Dyer conjecture “may be so difficult that we are in the wrong century” to unravel it.
R. Yes, absolutely. It is a really tough downside. With present methods it doesn’t appear that it may be addressed and it’s attainable that we must wait till the subsequent century or extra for it to be resolved.
P. In reality, Fermat’s Last Theorem is from 1637, with the handwritten annotation within the margin of a web page of a ebook by Diophantus.
R. Yes, Fermat wrote in a margin that he had the answer for that end result, however he was most likely incorrect and it took 300 years to really show that assertion.
P. If the precedent of Fermat’s Last Theorem have been repeated, to have the answer of the Birch and Swinnerton-Dyer conjecture we must wait till the 12 months 2323.
R. Could be, might be. You by no means know, there are issues which can be resolved sooner, others take longer. It is tough to foretell how lengthy it should take to resolve the issue.
In arithmetic, the cutting-edge instruments proceed to be paper and pen.
P. The Spanish mathematician Javier Gómez Serrano has teamed up with Google Deepmind to attempt to resolve the Navier-Stokes million-dollar downside within the subsequent 5 years. [relacionado con las ecuaciones que describen el movimiento de los fluidos, como el agua y el aire]. Do you employ synthetic intelligence with the Birch and Swinnerton-Dyer conjecture?
R. From what I perceive, they’re much extra superior with the Navier-Stokes downside and the prospects for fixing it quickly are higher. In the case of the Birch and Swinnerton-Dyer conjecture we’re nonetheless a lot additional behind. I personally do not use synthetic intelligence a lot. It might be helpful for trying up references for a given matter, however for doing analysis I would not say it is very helpful at this level.
P. What do you employ? Paper and pen?
R. Yes, to do analysis, usually, paper and pen.
P. It’s shocking. One may suppose that cutting-edge arithmetic is utilizing cutting-edge instruments.
P. In arithmetic it’s nonetheless paper and pen.
https://elpais.com/ciencia/2025-12-15/raul-alonso-investigador-las-matematicas-mas-hermosas-no-se-investigan-por-sus-aplicaciones-practicas.html