Credit … Ruth Fremson / The New York Times
Most of the best mathematicians discovered the subject when they were young, often excelling in international competitions.
By 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. “The math was remarkably mediocre, on average, that is, in some tests I did reasonably well. But other tests, I almost failed.”
As a teenager, Dr. Huh wanted to be a poet and spent a couple of years after high school pursuing this creative pursuit. But none of his writings were ever published. When he entered Seoul National University, he studied physics and astronomy and considered a career as a science journalist.
Looking back, he recognizes flashes of mathematical understanding. In high school in the 90s, he was playing a computer game, “The 11th Hour.” The game included a puzzle of four knights, two black and two white, placed on a small chessboard in a strange way.
The task was to exchange the positions of the black and white knights. He spent more than a week fidgeting before realizing that the key to the solution was to find in which squares the knights could move. The chess puzzle could be reformulated as a graphic where each knight can move to an unoccupied neighboring space and a solution could be seen more easily.
Reformulating mathematical problems by simplifying and translating them so that the solution is more obvious has been the key to many advances. “The two formulations are logically indistinguishable, but our intuition only works in one of them,” Dr. Huh.
A mathematical thinking puzzle
A mathematical thinking puzzle
Here is the puzzle that June Huh overcame:
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Objective: To exchange the positions of the black and white knights. →
A mathematical thinking puzzle
Many players stumble upon trial and error looking for a pattern. This is what Dr. Huh, and he almost gave up after hundreds of attempts.
He then realized that the strangely shaped board and L-shaped movements of the knights are irrelevant. What matters is the relationships between the squares.
Reformulating a problem into something easier to understand is often key for mathematicians to make progress.
A mathematical thinking puzzle
We number the squares so we can keep track of them.
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A mathematical thinking puzzle
Consider a gentleman in box 1. You can only move in box 5, while a gentleman in box 5 can move in 1 or 7.
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A mathematical thinking puzzle
This can be represented as a network diagram, what mathematicians call a graph. The lines indicate that a knight can move between squares 1 and 5 and between squares 5 and 7.
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A mathematical thinking puzzle
By strangely extending this analysis to the chessboard, we obtain this graph:
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Now, we can place the knights in this graph, the white knights in spaces 1 and 5, the black knights in spaces 7 and 9.
A mathematical thinking puzzle
The problem remains exchanging the positions of the black and white knights. For each move, a knight can slide to an adjacent empty node.
The reformulated version is much easier to figure out. Here is an answer:
A mathematical thinking puzzle
Using the graph with the numbered nodes as a decoder ring, we find the movements in the original board.
A mathematical thinking puzzle
Ruth Fremson / The New York Times
“Wind the same puzzle in this new way, which better reveals the essence of the problem, suddenly the solution was obvious,” said Dr. Huh. “That made me think what it means to understand something.”
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It was only in 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 a visiting professor at Seoul National.
Dr. Hironaka was giving a class on algebraic geometry, and Dr. Huh, long before I got a PhD, thinking I could write an article about Dr. Hironaka, attended. “It’s like a superstar in most of East Asia,” Dr. Huh said of Dr. Hironaka.
Initially, the course attracted more than 100 students, said Dr. Huh. But most of the students quickly found the material incomprehensible and left the class. Dr. Huh continued.
“After three lectures, we were like five,” he said.
Dr. Huh started having lunch with Dr. Hironaka to talk about math.
“It was mostly him talking to me,” Dr. Huh said, “and my goal was to pretend to understand something and react in the right way for the conversation to continue. It was a difficult task because I didn’t really know what was going on.” .
Dr. Huh graduated and began working on a master’s degree with Dr. Hironaka. In 2009, when Dr. Huh applied to a dozen graduate schools in the United States for a doctorate.
“I was pretty sure that despite all my failed math courses on my undergraduate transcript, I had an enthusiastic letter from a Fields medalist, so I would be accepted from many undergraduate schools.”
All but one rejected him: the University of Illinois Urbana-Champaign put him on a waiting list before finally accepting him.
“It was a very suspenseful few weeks,” Dr. Huh said.
In Illinois, he began work that led him to excel in the field of combinatorics, an area of mathematics that finds out how many ways things can be mixed. At first glance, he seems to be playing with Tinker Toys.
Think of a triangle, a simple geometric object — what mathematicians call a graph — with three edges and three vertices where the edges meet.
So, one can start asking questions like, given a certain number of colors, how many ways are there to paint the vertices where none can be the same color? The mathematical expression that gives the answer is called the chromatic polynomial.
More complex chromatic polynomials can be written for more complex geometric objects.
Using tools from his work with Dr. Hironaka, Dr. Huh proved 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, demonstrated Rota’s conjecture, which involved more abstract combinatorial objects known as matroids instead of triangles and other graphs.
For matroids, there is another set of polynomials, which exhibit behavior similar to chromatic polynomials.
His test incorporated an esoteric piece of algebraic geometry known as Hodge’s theory, named after William Vallance Douglas Hodge, a British mathematician.
But what Hodge had developed “was just one example of this mysterious and ubiquitous appearance of the same pattern in all mathematical disciplines,” Dr. Huh said. “The truth is, we, even the best experts in the field, don’t know what it really is.”