A Ukrainian mathematician who proved the best way to pack spheres in eight dimensions so they take up the least amount of space and an Oxford expert who solved conundrums in the distance between prime numbers are among the winners of the Fields Medal, considered for the equivalent of a Nobel Prize in mathematics.
The winners of the prize, presented at the International Mathematical Union awards ceremony in Helsinki, were announced as Prof. James Maynard, 37, of the University of Oxford, Prof. Marina Wiazowska, 37, of the École politechnique fédérale de Lausanne, Hugo Duminil -Kopen, 36, of the University of Geneva and Institut des Hautes Études Scientifiques, and Jun He, 39, of Princeton University.
While the first Fields Medal was awarded in 1936, there was a hiatus until 1950, when it was awarded every four years to up to four mathematicians who were under 40 years of age.
Vyazovska, who was born and raised in Kyiv, is only the second woman to receive the award, following the victory of Iranian mathematician Mariam Mirzakhani, who became the medalist in 2014. Mirzakhani died of breast cancer in 2017.
Speaking after receiving the 2018 New Horizons in Mathematics Award, Wiazowska said one thing she really likes about mathematics is that it’s always clear where the truth lies.
James Maynard: “Prime numbers are like atoms to mathematicians.” Photo: Ryan Cowan
“A theorem is always either wrong or right … if you’re taking a history exam and someone asks you ‘what were the causes of the French Revolution?’, how can you tell which answer is wrong and which is right?” So it’s always been very mysterious to me,” she said.
Speaking to the Guardian from his hotel room in Helsinki, Maynard – who is expecting his baby soon – said he learned of his win while climbing a ladder doing house repairs.
“I took out my phone to use as a flashlight to see if I messed up the picture or not. And I noticed that I had then received an email from the IMU president asking for a zoom call,” he said. “When I got that email, I suspected what it might mean.”
Maynard’s citation points to his “magnificent contributions to analytic number theory”—among them his work on the distribution of prime numbers.
“Prime numbers are like atoms to mathematicians,” Maynard said. “In the same way that you can understand a lot about chemicals by knowing the atoms that make them up, you can understand a lot about integers and how they interact with multiplication – which turns out to be very important for things like cryptography – if you understand things of the prime numbers.’
A key step in trying to understand prime numbers, Maynard said, is to look at the size of the gaps between them. “Unfortunately, we don’t understand this very well, despite the fact that it’s certainly been studied for hundreds of years and possibly thousands of years,” he said.
However, Maynard made a number of breakthroughs, including showing that sometimes prime numbers are unusually close together and sometimes unusually far apart.
Prof Andrew Granville, a former mentor, said that when Maynard made an early key discovery about how often pairs of prime numbers occur that are two steps apart – such as three and five – Graville told the young mathematician that he should he made a mistake. But Maynard was not.
“It was a real shock,” Granville said. “And the thing is, he’s not a one-horse wonder…James came close to one [question] after another and has just made tremendous progress.’
Granville also praised the work of Wiazowska, who solved the problem of the densest way to pack spheres in eight dimensions and, working with others, in 24 dimensions.
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As Granville notes, the conundrum originated in Elizabethan England, when Sir Walter Raleigh wondered how to calculate the number of cannonballs in a pile. This was resolved by Raleigh’s assistant Thomas Harriot, who then began to ponder how the spheres could be packed to take up the least amount of space. The answer, according to Renaissance astronomer Johannes Kepler, is a pyramidal pattern—like the one seen on an orange stand. However, his assumption was proven only in recent years.
Wiazowska, Granville said, took the question even further, finding the solution in higher dimensions. “It turns out that in dimensions eight and 24, the solution is much easier than our common dimension three,” Wiazowska said in 2018.
Maynard told the Guardian that winning the award was a great honor but surreal.
“It certainly feels a little strange to imagine my name on some list of great mathematicians when as a kid I read about a lot of these numbers and they kind of motivated me,” he said. “So putting my name on that list feels totally weird.”
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