Contents

- 1 A Puzzle of Mathematical Thinking
- 2 A Puzzle of Mathematical Thinking
- 3 A Puzzle of Mathematical Thinking
- 4 A Puzzle of Mathematical Thinking
- 5 A Puzzle of Mathematical Thinking
- 6 A Puzzle of Mathematical Thinking
- 7 A Puzzle of Mathematical Thinking
- 8 A Puzzle of Mathematical Thinking
- 9 A Puzzle of Mathematical Thinking
- 10 A Puzzle of Mathematical Thinking

Most top 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. “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, and he spent a couple of years after high school chasing that 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 insight. In middle school in the 1990s, 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, oddly shaped chess board.

The task was to exchange the positions of the black and white knights. He spent more than a week flailing before he realized the key to the solution was to find which squares the knights could move to. The chess puzzle could be recast as a graph where each knight can move to a neighboring unoccupied space, and a solution could be seen more easily.

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

## A Puzzle of Mathematical Thinking

## A Puzzle of Mathematical Thinking

Here is **the puzzle that June Huh beat**:

**Goal: **Exchange the positions of the black and white knights. →

It was only in his last year of college, when he was 23, that he discovered math again. 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 teaching a class about algebraic geometry, and Dr. Huh, long before receiving a Ph.D., thinking he could write an article about Dr. Hironaka, attended. “He’s like a superstar in most of East Asia,” Dr. Huh said of Dr. Hironaka.

Initially, the course attracted more than 100 students, Dr. Huh said. But most of the students quickly found the material incomprehensible and dropped the class. Dr. Huh continued.

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

Dr. Huh started getting lunch with Dr. Hironaka to discuss 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 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, Dr. Huh applied to about a dozen graduate schools in the United States to pursue a doctoral degree.

“I was fairly confident 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 grad 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.

At Illinois, he started the work that brought him to prominence in the field of combinatorics, an area of math that figures out the number of 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 start asking questions like, given a certain number of colors, how many ways are there to color the vertices with the rule that the vertices at each end of an edge cannot be 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, Dr. Huh proved Read’s conjecture, which described the mathematical properties of these chromatic polynomials.

In 2015, Dr. Huh, together with Eric Katz of Ohio State University and Karim Adiprasito of the Hebrew University of Jerusalem, proved the Rota Conjecture, which involved more abstract combinatorial objects known as matroids instead of triangles and other graphs.

For the matroids, there are another set of polynomials, which exhibit behavior similar to chromatic polynomials.

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

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

**Correction:**

July 5, 2022

Because of an editing error, this article misstated how chromatic polynomials are calculated. The two vertices at the end of an edge in a graph must be different colors, not that all of the vertices of the graph must be different colors.