Fields Medals in Mathematics won by four under 40

Credit…Ruth Fremson/The New York Times

Most top mathematicians discovered the subject when they were young, often excelling in international competitions.

In contrast, math was a weakness for June Huh, who was born in California and grew up in South Korea. “I was pretty good at most subjects except math,” he said. “Maths was on average remarkably mediocre, meaning I did quite well on some tests, but almost failed on others.”

As a teenager, Dr. Huh becoming a poet, and he spent a few years after high school pursuing that creative pursuit. But none of his writings were ever published. While attending Seoul National University, he studied physics and astronomy and considered a career as a science journalist.

Looking back, he recognizes flashes of mathematical insight. In high school in the 1990s, he played a computer game, “The 11th Hour.” The game featured a puzzle of four knights, two black and two white, placed on a small, oddly shaped chessboard.

The task was to exchange the positions of the black and white knights. He spent more than a week swinging before realizing that the key to the solution was finding which squares the knights could go to. The chess puzzle could be rearranged as a chart where each knight could move to a neighboring unoccupied square, and a solution could be seen more easily.

Rearranging math problems by simplifying and translating them in a way that makes a solution clearer has been the key to many breakthroughs. “The two formulations are logically indistinguishable, but our intuition works in only one of them,” said Dr. huh.

A puzzle of mathematical thinking

A puzzle of mathematical thinking

Here is the puzzle that June Huh hit

The New York Times

Goal: Swap the positions of the black and white knights.

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It wasn’t until his senior year of college, when he was 23, that he rediscovered mathematics. That year, Heisuke Hironaka, a Japanese mathematician who had won a Fields Medal in 1970, was visiting professor at Seoul National.

dr. Hironaka taught algebraic geometry, and Dr. Huh, long before he got his PhD, he thought he’d write an article about Dr. Hironaka could write. “He’s like a superstar in most of East Asia,” said Dr. Huh about Dr. Hironaka.

Initially, the course attracted more than 100 students, said Dr. huh. But most students soon found the material incomprehensible and dropped the lesson. dr. Huh continued.

“After three lectures there were five of us,” he said.

dr. Huh went to lunch with Dr. Hironaka to discuss mathematics.

“It was mostly him who talked to me,” said Dr. Huh, “and my goal was to pretend I understood something and respond in the right way so that the conversation kept going. It was a challenging task because I really didn’t know what was going on.”

dr. Huh graduated and started working on a master’s degree with Dr. Hironaka. In 2009, when Dr. Huh applied to a dozen graduate schools in the United States to earn a doctorate degree.

“I was pretty sure that despite all my failed math courses in my undergrad transcript, I had an enthusiastic letter from a Fields Medalist, so I would be accepted from many, many graduate schools.”

All but one rejected him – the University of Illinois Urbana-Champaign put him on a waiting list before finally accepting him.

“It’s been an exciting couple of weeks,” said Dr. huh.

In Illinois, he began the work that brought him fame in the field of combinatorics, a field of mathematics that finds out how many ways things can be shuffled. At first glance, it looks like playing with Tinker Toys.

Consider a triangle, a simple geometric object — what mathematicians call a graph — with three edges and three vertices where the edges meet.

One can then ask questions such as, given a certain number of colors, how many ways are there to color the vertices where none can have the same color? The mathematical expression that gives the answer is called a chromatic polynomial.

More complex chromatic polynomials can be written for more complex geometric objects.

Using tools from his work with Dr. Hironaka, proved Dr. Huh Read’s conjecture, which described the mathematical properties of these chromatic polynomials.

In 2015, Dr. Huh, along with Eric Katz of Ohio State University and Karim Adiprasito of the Hebrew University of Jerusalem, developed the Rota conjecture, which involved more abstract combinatorial objects known as matroids rather than triangles and other graphs.

For the matroids, there is another set of polynomials, which exhibit the same behavior as chromatic polynomials.

Their proof yielded an esoteric piece of algebraic geometry known as the Hodge theory, named after William Vallance Douglas Hodge, a British mathematician.

But what Hodge had developed “was just one example of this mysterious, ubiquitous appearance of the same pattern across all mathematical disciplines,” said Dr. huh. “The truth is that we, even the best experts in the field, don’t know what it really is.”

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