Lessons from June Huh: From High-School Dropout to Fields Medal Winner
My notes on June Huh: a High-School Dropout who Became a Fields Medal Winner and his teacher Heisuke Hironaka.
I couldn’t help but wonder into a great article about June Huh who happens to be a high-school drop out who has recently won the biggest prize in mathematics: a Fields Medal. Fields medals don’t get awarded based on politics nor small feats (like the Nobel peace prize) so I decided to take some notes on this story and highlight the parts which I thought were important.
Huh Didn’t Like School
School was excruciating for him. He loved to learn but couldn’t focus or absorb anything in a classroom setting. Instead, he preferred to read on his own — in elementary school, he devoured all 10 volumes of an encyclopedia about living things — and to explore a mountain near his family’s apartment.
He’s not the first to not love school.
“It is nothing short of a miracle that modern methods of instruction have not yet entirely strangled the holy curiosity of inquiry.” - Albert Einstein
An Extremely Late Bloomer
He wasn’t attracted to mathematics from the get-go and only developed a passion for it late in life.
He was indifferent to the subject, and he dropped out of high school to become a poet. It would take a chance encounter during his university years — and many moments of feeling lost — for him to find that mathematics held what he’d been looking for all along.
You Don’t Need an Amazing Work Ethic to be Exceptional
On any given day, Huh does about three hours of focused work. He might think about a math problem, or prepare to lecture a classroom of students, or schedule doctor’s appointments for his two sons. “Then I’m exhausted,” he said. “Doing something that’s valuable, meaningful, creative” — or a task that he doesn’t particularly want to do, like scheduling those appointments — “takes away a lot of your energy.”
In fact, it’s mentioned that there are periods where he doesn’t even perform the three hours of focused work that he sets out to perform and instead takes a break to re-read books which he encountered when he was younger.
No Set Deadlines nor Goal-Based Orientation
He finds that forcing himself to do something or defining a specific goal — even for something he enjoys — never works. It’s particularly difficult for him to move his attention from one thing to another. “I think intention and willpower … are highly overrated,” he said. “You rarely achieve anything with those things.”
This shows the great power of letting one’s mind wonder -- not forcing yourself to grind through what you may not want to do, albeit this may not work for everyone. Huh later in his life developed a huge passion for mathematics which I believe allowed him to unconsciously ‘work’ on some of the problems he encountered without necessarily knowing it.
On Rejection
Huh applied to about a dozen doctoral programs in the U.S, but because of his undistinguished undergraduate experience, he was rejected by all of them except for one, and the only reason he got into that one (University of Illinois) is due to a recommendation by his teacher and Fields medal winner Heisuke Hironaka.
Learning Driven by Creativity and Deep Understanding
Huh’s inadvertent proof of Read’s conjecture, and the way he combined singularity theory with graphs, could be seen as a product of his naïve approach to mathematics. He learned the subject mainly on his own and through informal study with Hironaka. People who have observed his rise over the last few years imagine that this experience left him less beholden to conventional wisdom about what kinds of mathematical approaches are worth trying. “If you look at mathematics as a kind of continent divided into countries, I think in June’s case nobody really told him there were all these borders. He’s definitely not constrained by any demarcations,” said Robbert Dijkgraaf, the director of IAS.
In other words, taking the unconventional approach sometimes can give you a huge advantage over the rest of the field or research area. Exploring things and learning about them using different methodologies can lead to many insights that others might not be able to realize or see.
Another great quote on how his unconventional approach and the pursuit of deeper understanding:
He proceeds just as deliberately when doing mathematics. Wang was shocked when he first witnessed it. “I have this math competition experience, that as a mathematician you have to be clever, you have to be fast,” he said. “But June is the opposite. … If you talk to him for five minutes about some calculus problem, you’d think this guy wouldn’t pass a qualifying exam. He’s very slow.” So slow, in fact, that at first Wang thought they were wasting a lot of time on easy problems they already understood. But then he realized that Huh was learning even seemingly simple concepts in a much deeper way — and in precisely the way that would later prove useful.
The Importance of Having a Routine
Huh’s entire life is built on routine. “Almost all of my days are exactly the same,” he said. “I have a very high tolerance for repetition.” He has trouble staying asleep and usually wakes up at around 3 a.m. He then goes to the gym, has breakfast with his wife and two sons (one is 8 years old, the other just turned 1), and walks his eldest to school before heading to his Princeton office.
Huh likes repetitive, mindless activities like cleaning, dishwashing and the physical act of transcribing what he reads into a notebook. He often works in the public library, in the children’s section, where it’s pretty noisy. “I don’t like quiet places,” he said. “It makes me sleepy.” Huh says this about many things.
He goes for a long walk after lunch each day, then returns to his office to do some more work (unless he’s already hit his three-hour quota) before heading home.
The Value of Great Teachers
It took Huh six years to graduate. In that sixth year, he enrolled in a class taught by the famed Japanese mathematician Heisuke Hironaka, who won the Fields Medal in 1970. Hironaka was charismatic and Huh quickly fell under his sway. He was vital in leading Huh into his mathematical adventure and late-life success.
Who is Heisuke Hironaka?
Heisuke Hironaka is one of the premier algebraic geometers of the twentieth century.
He is best known for his 1964 work on the resolution of singularities of algebraic varieties over a field of characteristic zero. Hironaka took a strikingly original approach and created new algebraic tools suited to the problem in order to solve it. These tools also proved useful for attacking other problems far removed from the original resolution of singularities quest which led to him developing the new tool-set in the first place.
Lessons from Hironaka
Hironaka has contributed much time and effort to encouraging young people interested in mathematics. In 1980, he started a summer seminar for Japanese high school students and later added one for Japanese and American college students; the seminars ran for more than two decades under his direction and continue to this day. Later on in his life, he wrote a best-selling memoir called The Joy of Learning which inspired a generation of Korean and Japanese students to learn and explore math.
Coming from my experience as a mathematician, I think that what’s interesting is talking and ideas, not well organized lectures where you sit and listen and take notes. When young people want to have a creative life, they should learn to enjoy talking about ideas, even if the ideas are not well formulated or keep changing. In fact, one of the most interesting and enjoyable parts of a creative activity is that ideas change. This is how the seminars are run.
On getting students to use their instinct and not just rely on knowledge:
When a person works, he must have knowledge or he will make terrible mistakes. But at the same time, knowledge alone doesn’t do anything new. You must have instinct and somehow be conscious of making use of instinct. It is an interesting question how to give kids knowledge without having them lose their instinctive power. If you just keep pounding them with knowledge, most lose their instinct and try to depend on knowledge. This balance between knowledge and instinct is interesting.
You can tell that Hironaka had a great passion for teaching and for letting his students carve their own path. He was known for making up his own theory as he went along rather than using existing theories; for using different approaches to tackling a problem rather than attempting to use a carved path and he left a great mark on Huh.
Hironaka didn’t necessarily leave a great impression on everyone though. Huh noted that although his class began with 200 students in it, only five students (Huh being one among them) were left after a few weeks passed. Although Hironaka’s unstructured and creative approach may not have attracted everyone, Huh loved it and noted that he loved going into every lecture with the suspense, freedom and uncertainty that each one brought, even though he might have not understood a lot of it.
“The math students dropped out because they could not understand anything. Of course, I didn’t understand anything either, but non-math students have a different standard of what it means to understand something,” Huh said. “I did understand some of the simple examples he showed in classes, and that was good enough for me.”
This wasn’t your average re-hashed material available in a normal college course. This was a great mathematician discussing how to tackle difficult problems he was struggling with right in-front of a live audience!
Huh discovered that this kind of mathematics could give him what poetry couldn’t — the ability to search for beauty outside himself in a way that opened him up even more than writing had. He found that unlike when he was writing, he was never motivated by the desire for recognition. He simply wanted to just do math.
More Lessons from Hironaka
When Hironaka was asked about when he got serious about the subject, he answered:
“I think one of the times when I seriously started thinking about the possibility of becoming a mathematician was in senior high school, when a mathematics professor from Hiroshima University came to my school. He gave a general lecture to the students. It was a bit technical, so I couldn’t understand everything. But he said at the beginning of his talk something like, “Mathematics is a mirror in which you can project everything in the world.” I was very puzzled by that, but also very impressed. I applied to Hiroshima University because I wanted to study with him.”
In fact, Hironaka initially went into physics due to being inspired by Hideki Yukawa (who was the first Nobel laureate from Japan) but later realized that he was more suited to mathematics (since mathematical problems gave him more excitement than physics based ones).
Hironaka (on working with the best and learning from peers) is quoted as saying:
“I tell youngsters, if you go abroad or even if you study in Japan, choose the best scholar in the field. But don’t expect you can learn from him! The amazing thing is that with that kind of person, there are many talented young people around, and you learn a lot from them.”
When asked on what attracted him to the resolution of singularities, he mentioned that although he studies quite a bit of abstract algebra – he found the field itself too abstract and that he was attracted to geometry (even though he might not have been able to develop a good intuition for geometries in higher dimensional spaces). On picking a technique to tackle a problem, he noted:
“I like basic things. Very clever people tend to jump to the new techniques: something is developing very fast, and you want to be on top of it; and if you are smart, you can be a top runner. But I am not so smart, so it is better that I start something where there are no techniques for the problem, and then I can just build step by step. But actually, it was not so hard. It turned out to be easier than I thought.”
On solving the problem and learning from Grothendieck:
“Let me explain a little bit about geometry. Geometry has global problems and local problems. Local problems are usually done by very concrete calculations. For instance, if you have an equation, then you can write down the equation, take its Taylor expansion, look at the terms, play with them. But then when you go back to the global problem, the local solutions do not fit each other. That is one of the problems that Zariski had. He had extremely local techniques: you have some geometric object, you modify it, and you localize it. If you localize it, then you can do many tricks, but then later you cannot connect it to have a global solution. With the resolution of singularities, Zariski had a hard time even in dimension 3, and finally he gave up. Generally speaking, it’s easy when you have one equation. But if you have many equations, then it’s difficult, or people had the impression that it’s difficult. But I observed that one can use induction to handle many equations. So I started from dimension 1, but with many equations. Then I noticed that the next dimension might have many equations, but it’s the same style. It’s a very simple observation, but that helped my local theory. Still, the global problem was there. You can’t have global coordinates; only locally do you have coordinates and equations. So I had a problem there, but Grothendieck—Grothendieck is an amazing fellow! He doesn’t look at the equations. He just looks at everything globally from the beginning.”
In other words – start simply and seek feedback from your peers (or from the best).
In fact, Huh used a similar approach in earning his fields medal. He did extremely important work on the Rota conjecture with two collaborators —Karim Adiprasito, a master at combinatorics from Hebrew University of Jerusalem and Eric Katz, a combinatorial algebraic geometry and arithmetic geometry specialist from Ohio State University and they were both vital in proving the conjecture.
Final Takeaways
It takes great dedication and passion to do what June Huh did, and you can get a glimpse of what a great teacher in this Numberphile video where they ask Huh to explain the g-conjecture and he does an amazing job in doing so while keeping things extremely concrete and simple:
Quanta Magazine did an amazing job in reporting this story and full credit goes to them for the above content. I simply wanted to highlight what notes I took while reading about Huh – so hopefully this helps inspire someone who may not have much confidence or may be starting late in math. Some key lessons which I learned by reading about both Huh and Hironaka are provided below:
It’s never too late to do what you’re passionate about and to succeed in it.
There’s nothing wrong with being unconventional and seeking different paths.
Have ambitions and seek out difficult problems to solve.
When solving problems, don’t be afraid to explore and invent – seek a deeper understanding and look for unexplored pathways and problems which haven’t been probed.
Last but not least: seek out the best and learn and work with them.
Huh accomplished what many people would only have dreamed of. There are many more lessons and takeaways that can be both from him and his amazing teacher Heisuke Hironaka – here I simply outlined what I found important. If there’s anything I missed, feel free to leave a comment! Thank you for reading!
References:
https://www.quantamagazine.org/june-huh-high-school-dropout-wins-the-fields-medal-20220705/
https://www.quantamagazine.org/a-path-less-taken-to-the-peak-of-the-math-world-20170627/
https://www.daviddarling.info/encyclopedia/A/algebraic_geometry.html