Amazing Mathematical Breakthroughs Are Recognized by Field Medals

By: Jovia Zhang

Every four years, rare and prestigious Fields Medals are bestowed upon some of the most skillful and gifted mathematicians under the age of forty. Of the four recent recipients, one was the second woman ever to receive such an honor and one wasn’t even interested in the field of mathematics until he was 23.

June Huh, 39, of Princeton University; 37-year-old Maryna Viazovska of the Swiss Federal Institute of Technology; 35-year-old James Maynard of the University of Oxford; and 36-year-old Hugo Dumini-Copin of the Institut des Houtes Études Scietifiques and the University of Geneva were all proud winners of the 14-karat-gold medals, given out during a ceremony in Helsinki on Tuesday, July 5th.

According to the New York Times, “The Fields Medals, first awarded in 1936, were conceived by John Charles Fields, a Canadian mathematician. They and the Abacus Medal are unusual among top academic honors in that they go to people who are still early in their careers — younger than 40 years [old] on Jan. 1 — and honor not just past achievements but also the promise of future breakthroughs.”

Awarding the Fields every four years makes them even more significant, like an Olympic gold medal. Other than the Fields and Abacus awards, there is also another award for mathematicians: the Abel Prize. This award is modeled on the Nobel Prize in that it annually recognizes mathematicians for work during their careers.

From a chess puzzle to the packing of eight-dimensional spheres, here’s why each of these four recipients was awarded the Fields Medal, along with the mathematical discoveries each person made.

June Huh: Picking Up Math Again

Unlike most top mathematicians, June Huh, who was born in California and grew up in South Korea, did not excel at math from a young age. In fact, it was one of his worst subjects: “I was pretty good at most subjects except math,” he said. “Math was notably mediocre, on average, meaning on some tests I did reasonably OK. But other tests, I nearly failed.”

As a teenager, Dr. Huh wanted to be a poet. However, after spending a couple of years going down that creative path in high school—and publishing no poetry—he changed ideas and went in pursuit of a career as a science journalist.

Back when he was in middle school during the 1990s, the teenaged Huh used to play a computer game called “The 11th Hour.” The game included a puzzle that had two black and two white knight chess pieces on an oddly-shaped board.

The goal was to exchange the positions of the black and white knights. After spending more than a week failing at this head-scratching puzzle, Huh finally realized something: the key to solving the puzzle was figuring out which squares each knight could move to. The young man reassembled the puzzle as a graph showing nearby empty spaces where each of the four knights could move. His graph allowed him to clearly see the solution.

Recasting math problems by making them easier and rewriting them in a way that makes the answer clearer has been the key to many breakthroughs.

It was only during his last year of college that Huh picked up math again. After taking an advanced class taught by Heisuke Hironaka, a Japanese mathematician who had won a Fields Medal in 1970, Huh finally graduated and started working on a master’s degree. Eventually, he entered the University of Illinois Urbana Champaign and went on to discover and prove many geometrical theories, including the Rota Conjecture, which replaced triangles and other geometric shapes with matroids, a variety of abstract combinatorial objects.

Maryna Viazovska: High-Dimension Equivalents of Stacking Spheres

Of the 60 people who have received Fields Medals before this year, 59 were men. Stanford mathematician Maryam Mirzakhani, who won the award in 2014, was the first and, until now, the only woman to ever receive the award. “I feel sad that I’m only the second woman,” Dr. Viazovska said. “But why is that? I don’t know. I hope it will change in the future.”

Maryna Viazovska is a Ukrainian professor at the Swiss Federal Institute of Technology in Lausanne. She is known for proofs of higher-dimension equivalents of the packing of equal-sized spheres. According to the New York Times, “Dr. Viazovska’s work is a variation of a conjecture by Johannes Kepler more than 400 years ago. Kepler is best known for realizing that the planets move around the sun in elliptical orbits, but he also considered the stacking of cannonballs, asserting that the usual pyramid stacking was the densest way that they could be arranged, filling up just under 75 percent of the available space.”

This statement couldn’t be proved until 1998, when Thomas Hales of the University of Michigan succeeded in proving it with the help of a computer program.

So far, proving something similar for the packing of equal-sized spheres in dimensions higher than three has been impossible – with a few exceptions.

In 2016, Dr. Viazovska found the answer in eight dimensions. Within a week, with four other mathematicians, she proved that an arrangement known as the “Leech Lattice” was the best possible packing in 24 dimensions.

James Maynard: The Gap in Prime Numbers

A prime number is a whole number with only two divisors: one and itself. “I personally find them just totally fascinating,” said James Maynard of the University of Oxford. “They feel like some of the most basic and fundamental objects in mathematics.”

Although Euclid proved, more than 2,000 years ago, that there are an infinite number of prime numbers, questions about them still remain.

According to the New York Times, “One of the unsolved problems is the Twin Prime Conjecture. Other than 2, even numbers are not prime because all even numbers can be divided by two. Thus, other than 2 and 3, the smallest difference between two prime numbers can be 2, and such pairs, like 5 and 7 or 11 and 13, are called “twin primes.”

Even though as the numbers get larger the numbers of prime numbers (and twin primes) get scarcer, mathematicians still believe there’s an infinite number of twin primes. This is the problem Dr. Maynard has been working on for years.

In 2013, Yitang Zhang of the University of New Hampshire published a breakthrough in the problem – that there’s an infinite number of twin prime pairs whose separation is less than 70 million.

Six months later, Dr. Maynard managed to narrow that gap down to 600. Since then, a group of mathematicians narrowed Maynard’s gap down to 246. Dr. Maynard also proved that there are an infinite number of primes without the digit 7 in them. The same works for other digits, as well.

Dr. Maynard believes that someone who remembers “some high school math” can understand the problems with prime numbers.

Hugo Duminil-Copin: Magnetism, Physics and Math

The types of mathematics problems Hugo Duminil-Copin likes to work on are not entirely abstract. “Actually, for a long time, I was torn apart between physics and math,” he said. “I loved the idea of physics, that you want to describe the world around you. And at the same time, I loved the beauty of the math solution, where in some sense you have the concrete truth and there is no question about whether it is complete or not. It’s just a proof.”

Dr. Duminil-Copin’s most remarkable discovery involves studying the phase transition – as when ice melts into water – in a model of magnetism.

With what is known as the Ising model (named after Ernest Ising, a German physicist who created the model in 1924), physicists have studied ferromagnetism – materials whose atoms act like tiny bar magnets – and the phase transition from nonmagnetic to magnetic.

Although Dr. Duminil-Copin and his collaborators are still working on different problems with the model, they have proved that the phase transition is continuous.

According to the New York Times, “The breakthrough came by making an unexpected connection between the Ising model and percolation — how water moves through porous rocks, for instance. The mathematicians had come up with a broader generalization of percolation.”

The mathematicians were able to come up with an innovation by combining the two theories (the percolation theory and the Ising model).

Making mathematical breakthroughs in subject areas ranging from prime numbers to magnetism, these four mathematicians have truly proved themselves worthy of the prestigious Fields Medal.

Sources:https://www.nytimes.com/live/2022/07/05/science/fields-medal-math - hugo-duminil-copin-and-his-magnetic-math