June Huh, High School Dropout, Wins the Fields Medal

June Huh finds it hard to find his way. He is a professor in the mathematics department at the university. He paused every so often as he considered a fork in the path ahead as he made his way through the woods on this particular day in May. A pair of frog, a red-crested bird, a turtle, and a quick-footed fox are some of the animals he sees over the next two hours.

He says he is very good at finding things. One of my special abilities is that.

Huh was awarded the Fields Medal, the highest honor in mathematics, for his ability to wander through mathematical landscapes and find just the right objects, which he then uses to get the seemingly disparate fields of geometry and combinatorics to talk to each other in new and exciting. He has solved several major problems in combinatorics by forging a circuitous route through other branches of math to get to the heart of each proof. It's a little miracle when you find that path.

It was a path that was characterized by a lot of wandering and small miracles. He didn't want to be a mathematician when he was younger. He dropped out of high school to become a poet because he was not interested in the topic. It would take a chance encounter during his university years to find what he had been looking for.

His mathematical breakthrough was made possible by that poetic detour. His colleagues say that his artistry is evident in the way he discovers just-right objects at the center of his work and in the way he seeks a deeper significance in everything he does. Federico Ardila-Mantilla, a mathematician at San Francisco State University, said that mathematicians are similar to artists in that they look for beauty. It is really pronounced in his case. I enjoy his taste. He makes pretty things.

Ardila was surprised to learn that he came to mathematics after poetry.

He draws parallels between the two. He said it felt like you were grabbing something that was already there rather than thinking about it.

The Dropout

Huh does about three hours of work on a daily basis. He could think about a math problem, lecture to a class of students, or schedule doctor's appointments for his two sons. He said that he would be exhausted after that. It takes away a lot of your energy to do something that you don't want to do.

He doesn't have a lot of control over what he focuses on in those three hours. He didn't do anything for a few months in the spring. He wanted to revisit some of the books he had first encountered when he was younger, and so he did. Huh said that he didn't do anything. That is sort of a problem. Since then, he has made peace with this constraint. I tried to resist, but finally gave up on those temptations. I was better at ignoring deadlines.

He doesn't like being forced to do something or define a goal that he doesn't enjoy. It is hard for him to concentrate on one thing at a time. He thinks intention and willpower are not worth much. You don't achieve much with those things.

Since he was a young man, this has been the case. His parents were graduating from graduate school when he was born. The family moved to South Korea when Huh was young. His father taught statistics and his mother spoke Russian.

It was hard for him to go to school. He couldn't concentrate in a classroom. He preferred to read on his own and explore a mountain with his family. He got lost one time and ended up in an area that was restricted due to the presence of land mines.

He tried not to use math. Huh would copy the solutions from the back when his father tried to teach him how to use a workbooks. When his father ripped the pages out, Huh went to a bookstore and wrote down the answers. Huh said that he gave up at that time.

He dropped out of high school in the middle of his first year because he wanted to be a poet. He was very much in love. He said he cried after listening to good music. He talked about nature and his own experiences. He wanted to finish his work in two years so he wouldn't have to go to university. He chuckled and said that that didn't happen.

The exploration of the writing process was often painful and depressing for him. He wanted to be a great poet. I didn't want to be a great poet. He sees that version of himself as completely different.

He felt lost when he joined the university. He majors in astronomy and physics after flirting with the idea of being a science writer. He missed a lot of class and had to take more courses. He stated that he was lost. I don't know what I want to do. I wasn't sure what I was good at.

He found out that he was good at math after all.

True Beauty

Six years is a long time to graduate. He was in a class taught by the renowned Japanese mathematician Heisuke Hironaka, who won the Fields medal in 1970. Huh fell under Hironaka's influence.

Huh was attracted to his teacher on the first day of class. The math itself was also part of the equation. The course was designed to introduce students to the study of solutions to algebraic equations and their geometric properties. Hironaka taught his own work in the area of singularity theory, which deals with certain types of spaces. Huh said that he lectured about a very specific problem and proof that weren't necessarily correct. A few weeks later, only five students remained, Huh among them.

He had never seen research mathematics unfold in real time. The answers already worked out in Hironaka's lectures, which weren't polished like other undergrad courses. The act of trying to do something no one really knew how to do, and the freedom that came with not knowing, were Huh's favorite parts of it. Over the course of hundreds of years, the material taught in college has been updated. That is very different from what you see in front of your eyes.

This kind of mathematics gave Huh the ability to look for beauty outside himself, to try to grasp something external, objective and true, in a way that opened him up more than writing ever had. He said you don't think about yourself. There is no place for egos. He wasn't motivated by the desire to be recognized when he was a poet. He didn't want to do anything else.

Hironaka took him under his wings. Huh and Hironaka spent a lot of time together after Huh graduated from the university. He followed the professor back to Japan and stayed with him in Tokyo and Kyoto, as well as sharing meals and carrying his bags.

An Unexpected Discovery

He was rejected by all of them because of his undistinguished undergraduate experience. After starting his studies at the University of Illinois, he transferred to the University of Michigan to complete his PhD.

Huh cherishes his experiences in graduate school despite the challenges of living in a new country. He enjoyed the freedom of exploration that drew him to the subject in the first place, and he was able to dedicate himself entirely to math.

The man stood out. He was a graduate student in Illinois when he was able to prove a hypothesis in graph theory. In its simplest form, the problem is related to the equations n 4 + 5n 3 + 6n 2 + 3n + 1 which are attached to graphs. Let's say you want to make sure that no one has the same color in the graph. There are a lot of ways to color the graph. The total number of possibilities can be calculated using an equation that takes into account the number of colors being used.

They are unimodal because they increase and decrease at the same time. You can take the previous example. A unimodal sequence is formed by the values of its coefficients. There is also a sequence that islog conjugate. The square of the middle number is larger than the product of the terms on either side of it. For example, 6 2 5 3 is used.

Calculating these properties wasn't easy. Huh came out of the blue.

He had studied with Hironaka. The main objects of study in that field are called algebraic varieties. Huh only knew about the log concave numbers associated with certain types of algebraic varieties because of his studies. Huh wanted to find a way to make a variety with the coefficients of the graph from the original question.

The solution he came up with shocked the math community. The University of Michigan accepted him to their graduate program after rejecting his initial application.

The achievement of Huh was impressive not only because he had solved the problem, but also because it seemed impossible for so long. He showed that there was more to graphs than they showed.

He was also liked by mathematicians. In speaking with him, it was clear that he was thinking both deeply and broadly about the concepts he was working with, as his talks were always accessible and concrete. Matthew Baker is a mathematician at the Georgia Institute of Technology. After Baker met him for the first time, he wondered who he was.

Mircea Musta is Huh's adviser at the University of Michigan. He had a program in mind and ideas about how to pursue it. Musta said that he was more like a coworker. He already had an idea of what he was looking at.

He is humble and down-to-earth according to many of his colleagues. It didn't feel good when he learned he had won the Fields medal. Even though you are happy, you are worried that they will eventually figure out that you are not that good. I am a reasonably good mathematician.

Escape From Space

Graphs are just one type of object that can be defined as matroids. A point on a two-dimensional plane is considered. You can say that if more than two points are on a line, they are dependent. Matroids are objects that capture notions of dependence and independence in a variety of contexts.

Similar to graphs, there are equations attached to matroids. It was thought that the coefficients of the polynomials for the more general objects should be log-concave. The techniques Huh used were only used to show log concavity for a very narrow class of matroids.

Huh was able to broaden the class of matroids with the help of the mathematician. The recipe they followed was similar. The strategy was to start with the object of interest and then use it to make a variety. Some of the properties of a cohomology ring can be used to prove log concavity.

One problem was present. The majority of matroids don't have any kind of geometric foundation, which means there isn't an algebraic variety to associate them with. They figured out a way to write down the ring directly from the matroid. They used a new set of techniques to show that it behaved as if it had come from an actual type. They proved log concavity for all matroids in order to solve the problem of Rota's conjecture once and for all. Baker said it was remarkable that it worked.

The work shows that you don't need a lot of room to do geometry. It made me rethink what geometry is. He was able to develop an even broader range of methods as a result of that idea.

Building the right cohomology ring requires a lot of work and a lot of uncertainty. It was an aspect of the work that Huh liked the most. He said there was no guiding principle. It's just a matter of making a guess.

He functions best in his day-to- day life because of that lack of intention. It was like he had found a mathematical program that was perfect for him. He said that things just happen by themselves.

The Heart of Things

Huh speaks slowly, pausing often and choosing his words carefully, and carries himself in a calm, peaceful manner. Botong Wang, a mathematician at the University of Wisconsin, Madison, has collaborated with Huh on a number of important recent results.

When doing mathematics, he proceeds the same way. When he saw it for the first time, he was shocked. He said that as a mathematician you have to be quick. You would think this guy wouldn't pass a qualification exam if you talked to him for five minutes. The man is very slow. It was so slow that Wang thought they were wasting a lot of time on easy problems. Huh was learning seemingly simple concepts in a much deeper way, and that would prove useful.

Graham Denham, a mathematician at Western University in Ontario, said that June likes to do things in the right way.

When Huh said they should take some more time to find a cleaner, more appealing approach, they had just finished a 50-page proof of a problem closely related to Rota's conjecture. He thought there was a better explanation out there. We were going to chuck that, then, shall we? "I'm not sure what you're talking about."

Two years is a long time to craft a better argument. Ardila said it was good that we were all tenured. The extra work was worth it. Ardila said that their end result was completely different and deeper.

Huh's mathematical work isn't the only one that this approach applies to. He wanted to become a cook. He used to make the same dish every day until it was perfect. He did that for six months. Kim says that the only dish he knows how to cook is that one.

Huh's life is built on a daily basis. He said that almost all of his days were the same. I have a high tolerance for repeating things. He tends to wake up at 3 a.m. He goes to the gym, eats breakfast with his wife and two sons, and walks his oldest to school before going to his office.

The office isn't occupied. He doesn't actually know how to do yoga, but there is a yoga mat on the floor and a large desk. There are a few stacks of papers neatly arranged on a shelf. There is a vacuum cleaner in this picture. Huh likes to clean, cook, and write in a notebook.

He works in the children's section of the library. He dislikes quiet places. It causes me to sleep. This is about a lot of things.

He goes for a long walk after lunch and then returns to his office to finish his work. He spends the rest of the night with his family and they all go to sleep in the same bed at 9 pm.

Sometimes this preference for routine can manifest in extreme ways. Huh said that when he completed his doctorate in Michigan, he would cut off most of the other things. He was not prepared for the winter when he moved to Ann arbor. He didn't have a lot of things, but he did need a blanket. He couldn't find a way to get to the mall. He said it was past his level of tolerance. I wanted to figure out how to go from here to there. He went to a nearby drugstore and bought 10 squares of fabric and a giant stapler to make a blanket.

He didn't want to do anything else. He describes that time as being almost monastic. He only spoke with Musta once a week.

Kim remembers visiting Huh when he was still in Illinois. I don't know if I should get married to him. He can't handle real life skills.

She got married to him in the year 2014). The Institute for Advanced Study is where they started working. Kim had to rely on Huh to get things done because she was not comfortable taking care of certain tasks in English. He stated that she was disappointed.

Their first child, Dan, was born later that year. She caught Huh doing math.

He said his wife is a lot more balanced than he is. Life has many aspects, and math is a very small one.

Kim claimed to be a real worker. He is a person of thought.

Huh has changed a lot since then. Huh said that he learned how to live a more balanced life after raising Dan. The period was a turning point. He spends a lot of time with Dan, drawing with him, and taking him to the bookstore and other places. He takes care of the logistical tasks that Kim wants him to do. He said, "We cannot just live with stapled blankets."

He is able to step away from math. He can take a break when something else requires him to and his mind no longer returns to working on problems when he is in an inactive state.

Kim said that he is a completely different person.

Heavy on Top

Some things haven't changed Huh can only work for a short period of time. Kim said that other people work one hour and just take a break. He said to focus for five minutes, 10 minutes.

He still searches for beauty. He comes back to questions about log concavity or similar concepts as a way to find that beauty.

He and others recently proved a fundamental problem about configurations of points, lines and planes called the Dowling-Wilson "top- heavy" conjecture. In a plane, every pair of points is connected by a line. If all the points are located on one line, the number of lines must always be greater than the number of points. Four points are placed at the corners of a square. Up to six lines are added to the square by tracing out the square and connecting opposite corners.

This idea is generalized by the top- heavy conjecture. The points are in some high-dimensional space. Consider the lines that connect pairs of points, the planes spanned by sets of three points, the three-dimensional subspaces, and so on. Think about the number of points, the number of lines, and the number of planes in a sequence. The first and last numbers are in a symmetrical position. The sequence is top heavy and the number corresponding to the higher-dimensional space will be as large as possible. The first half of the sequence has been shown to be unimodal by Huh and Wang.

Huh and Wang had to push his program further in order to adapt his ideas. They were working with matroids. The places where a space looks different when you zoom in on it are involved singularities. It was more difficult to prove certain properties of the cohomology rings because they had to build them from the matroids.

Huh began investigating a way to complete the break from geometry after they solved the problem for five years. It took him a long time to build the exact cohomology that a problem needed. Once that cohomology is found, mathematicians have to prove that it complies with certain properties, which can take a long time.

He and the mathematician were able to create a new theory that was completely different from the others. It allowed them to solve a problem called the strong Mason conjecture, which asked questions about the number of independent sets in matroids, and other mathematicians have already used it to re-prove Rota's conjecture. It opens the door to finding new problems, hints at an even deeper explanation for why all these log concavity statements are true, and intersects with problems in theoretical computer science that are just beginning to be explored.

Click of the Connection

There is something going on when Huh works. He can't remember how or when his ideas came to him. He doesn't have a lot of time to think. He said that at some point you realize that you know this. Maybe last week he didn't understand something, but now the pieces have clicked into place without his knowledge. It is similar to how your mind can surprise you when you are dreaming. He said it was amazing what human minds could do. It's nice to admit that we don't know what's happening.

This could also speak to the artist in him. He wants to find connections between different areas of math.

As a graduate student he followed the vision of the original program. It will be fascinating to see what the limits are.

Huh hasn't hit them yet. He will make beautiful things, according to mathematicians.

He shrugged when asked if he would ever try to write poetry again. It's possible, maybe. He didn't know. I am very interested in something else.