Fields medals in mathematics won by four under the age of 40

Most of the great mathematicians discovered this topic when they were young, and they often excel in international competitions.

By contrast, mathematics was a weak point for June Huh, who was born in California and raised in South Korea. “I was good at most of the subjects except for math,” he said. “The math was remarkably mediocre, on average, which means that on some tests I performed reasonably well but on others, I almost failed.”

As a teenager, Dr. Huh wanted to be a poet, and he spent two years after high school chasing this creative pursuit. But none of his writings were ever published. When he attended 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 ’90s, he was playing a computer game, “The 11th Hour”. The game featured a puzzle of four knights, two of them black and two white, placed on an oddly small chessboard.

The task was to exchange the positions of the black and white knights. He spent more than a week falling over before realizing that the key was to find the squares the knights could move to. The chess puzzle can be reformulated as a graph where each knight can move to an adjacent unoccupied space, and the solution can be seen more easily.

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

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Mathematical thinking puzzle

Mathematical thinking puzzle

over here Puzzle beat Jun huh:

Goal: The black and white knights exchange positions. →

Mathematical thinking puzzle

Many players stumble through trial and error in search of a pattern. This is what Dr. Huo did, and He could have given up After hundreds of attempts.

Then he realized that the single shaped board and the L-shaped knights’ movements were irrelevant. What matters are the relationships between the squares.

paraphrase a problem in Something easier to understand It is often a key for mathematicians to make breakthroughs.

Mathematical thinking puzzle

We Invited Square numbering so we can track it.

Mathematical thinking puzzle

I recommend Knight on square 1. He can only move to square 5 while the knight with number 5 can move to 1 or 7.

Mathematical thinking puzzle

This could be represented as a network diagram What mathematicians call a graph. The lines indicate that the rider can cycle between squares 1 and 5 and between squares 5 and 7.

Mathematical thinking puzzle

Expand this analysis to include a single shaped chessboard This graph is produced:

Now, we can put the jockeys on this graph, the white jockeys at spaces 1 and 5, and the black jockeys at 7 and 9.

Mathematical thinking puzzle

The problem is still the exchange of positions of black and white knights. For each movement, the rider can slide into an adjacent empty node.

The reworked version is Much easier to find out. Here is one answer:

Mathematical thinking puzzle

Use the graph with Numbered knots as a jaw ringthen we find the movements on the original board.

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Mathematical thinking puzzle

Presenting the same puzzle in this new way, which better reveals the essence of the problem, Suddenly the solution was clearDr. Huo said. “It got me thinking about what it means to understand something.”

He only discovered mathematics again in his last year of college, when he was 23 years old. That year, Hisuki Hironaka, the Japanese mathematician who won the Fields Medal in 1970, was a visiting professor at Seoul National University.

Dr. Hironaka was teaching a class on algebraic geometry, and Dr. Huh attended, long before he received his Ph.D., thinking he could write an essay on Dr. Hironaka. “He’s like a star in most of East Asia,” Dr. Huh said of Dr. Hironaka.

Dr. Huh said that the course initially attracted more than 100 students. But soon most of the students found the material incomprehensible and dropped the class. Dr. Huo continued.

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

Dr. Huo started having lunch with Dr. Hironaka to discuss mathematics.

“He was talking to me mostly,” said Dr. Huh, “and my goal was to pretend to understand something and respond in the right way so that the conversation would go on. It was a difficult task because I didn’t really know what was going on.”

Dr. Huh graduated and began working on his master’s degree with Dr. Hironaka. In 2009, when Dr. Huh applied to about a dozen top schools in the United States for his Ph.D.

“I was fairly confident that despite all the failed math courses in my undergraduate transcript, I had received an enthusiastic letter from a Fields Medalist, so I would be accepted from 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 a very interesting few weeks,” said Dr. Huh.

In Illinois, he began work that brought him to the fore in the field of combinations, the field of mathematics that quantifies the number of ways things can be mixed. At first glance, it looks like 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.

One can then start asking questions like, given a given number of colours, how many ways are there to color the vertices since none of them can be the same colour? 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. Ho proved Reed’s conjecture, which described the mathematical properties of these chromatic boundary modifiers.

In 2015, Dr. Huh, along with Eric Katz of The Ohio State University and Karim Adepracito of the Hebrew University of Jerusalem, demonstrated Rota’s theory, which involved more abstract combinatorial objects known as matroids rather than triangles and other graphs.

For matroids, there is another group of polynomials, which show behavior similar to chromatic polynomials.

Their proof was pulled into a mystical piece of algebraic geometry known as Hodge’s theorem, named after William Vallance Douglas Hodge, a British mathematician.

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