Hugo Duminil-Copin Wins the Fields Medal

When voices boom down the hall from the office of Hugo Duminil-Copin, the math department at the University ofGeneva is usually quiet. The department at the University of Fribourg and Duminil-Copin was known for shouting at each other. When their voices reached a peak, a professor next to them would have to close the door. Duminil-Copin said that they were hated by their coworkers.

He wouldn't make any changes about it. The heated discussions are important to how he does mathematics. He likes to share his ideas with others and put them back together as a group. A mathematician at the Swiss Federal Institute of Technology Zurich said that his way of doing math is very sporty.

It makes sense that Duminil- Copin is a sporty guy. He hikes, bikes, swims and climbs, and sometimes he gets a burst of mathematical inspiration. His interest in a diverse range of activities also characterizes his work, where he has samples of tools from various fields in an ongoing effort to transform mathematicians' understanding of phase transitions

The Fields Medal is the highest honor in mathematics. He studies models of fluids flowing through a porous medium like water flowing through coffee grounds. The spread of a disease, the circulation of a rumor or the advance of a forest fire can be represented by such models.

The study of these models was stuck before Duminil-Copin came on the scene. A relatively simple type of percolation model was well understood but not all models were. Duminil-Copin has been studying the more complex models since he was a graduate student.

You don't recognize it after he goes through this field. Everything is simpler and quicker. The results are getting better.

Competitions Won and Lost

People notice Duminil-Copin's enthusiasm the first time they see him. The laudatory lecture on Duminil-Copin's work will be delivered by Martin Hairer who is a mathematician at Imperial College London. Manolescu said yes. He likes to put his hands in it and do all the messy work, and even rewrite old proof to make sure he understands them properly. Hugo is very positive. He would like things to work.

In everything he does, that energy, passion and optimism is visible. He can't seem to sit still. While chatting in his office, he shifts in his seat, stands up and leans against the doorjamb, hands in pockets, or the arm of his chair, before sitting down. In less than two hours, he will drink two cups of coffee. He said that the coffee consumption became worse over time.

He's been active his whole life. Growing up in a suburb an hour's train ride from Paris, he spent his time playing guitar, hanging out with friends, and going camping, hiking, and cross-country skiing with his family.

He liked to play sports. He was able to test out different strategies and focus on something physical. His mother was a dancer before she became a teacher. He was interested in a lot of things, including math. He was good at it, but he didn't do as well as his peers. He played a sport that is popular in Europe, but he was hesitant about going to a specialized high school because he wanted to do more research. I didn't think I was worse at sports than math.

Duminil-Copin went to a high school in Paris that was geared toward math and science because he was good at it. He remembers being one of the worst students in his first year and then being placed in a class that only had the best students in it. He thought that would be a long year. He went back up in the ranks. He bombed on the qualification exam for the International Mathematical Olympiad, a math competition for high schoolers that Fields medalists have excelled at. He doesn't have an amazing problem-solving ability He wanted to be creative and work hard.

I never lose hope. He is positive that he will enjoy what he is doing.

He was able to take the national entrance exams for specialized universities when he was in the classe préparatoire. Manolescu met Duminil-Copin while they were together. He was more calm.

Duminil-Copin said that all of this added up to a normal childhood. A lot of people don't train for research because it's difficult for the mind. That's how I feel.

The Flow

The cole Normale Supérieure in Paris is a top university in France.

He didn't know if he would study math or physics. He said that he was in between wanting to understand the world but also wanting to know what was going on. He loved physics, but he realized that he wasn't satisfied with explanations unless they were fully rigorous, capable of saying that something was truly known. He said that with the right proof you have a sense of completion. It was difficult to find that with physics.

He focused on an area of math that was very close to physics and was motivated almost entirely by physical processes. Percolation was intuitive for him. "My first love"

Imagine an infinite grid of points that are connected by edges. flip a weighted coin The edge of the coin should be black if it lands on heads. Leave it alone if necessary. The fluid can flow through that part of the system if it is black. The mathematicians want to understand how clusters of connected black edges can come together.

A phase transition occurs when the probability of any two points connecting by a black edge increases. The behavior of the system changes in a similar way that water does when the temperature drops. Most systems go through a phase transition. If you increase the probability that the coin will land on heads, the system will pass a critical value, which means there is almost no chance that there will be an infinitely long connected component. Our fluid will flow through the system if it is above it.

The mathematicians want to understand how the system behaves at certain points. The simplest model of percolation was pinned down in percolation theory. Things began to change when Duminil-Copin decided to extend that understanding to other percolation models.

It all began with a swim and an idea that didn't work.

He had been trying to find a solution to the percolation problem for a long time. While he pondered over the problem, he came up with a different idea. It wouldn't work for the question he was trying to answer, but it was what he needed to solve a problem in a different field.

He said that he had a complete proof of the other problem when he was swimming in the sea.

He would immediately become a professor because of his PhD result. The hexagonal lattice is a graph of edges and is similar to a honeycomb. They wanted to know how many self-avoiding walks there were. The two mathematicians proved that there is a number of steps in the path. The Annals of Mathematics is considered the top journal in the field of mathematics.

The work done by the graduate students made it clear that he was not a normal student. He was offered a faculty position at the university on the night of his PhD defense. The Institute of Advanced Scientific Studies is in France. He was a full professor by the time he was NationMaster NationMaster.

Controlled Doubt

Duminil-Copin considers himself allergic to academic hierarchy despite his early success. He treated Peers as equals even though he became a postdoc.

There are only a few papers where Duminil-Copin is the only author. He would like to see the Fields Medal and other awards be joint. He said sharing it with someone made it more enjoyable. I can't remember the number of times I went to the office of one of my colleagues to explain a bad idea. He talks about everything, not just mathematics, with his friends and partners. It makes sense of things.

Manolescu said that he was one of the close friends and associates of the man. I would work with him for 5 hours. I would leave his office and die. He would take a coffee and have someone else talk about something else.

It was similar to having a good partner to climb a mountain with, according to one of Duminil-Copin's friends. He likened it to when you meet your soul mate. We created our own language over time. They don't have to say anything sometimes. A smile is all that's needed.

Tassion remembers jumping into each other's arms after finishing a proof with Duminil-Copin that both were excited about. Tassion said that math is something intimate. You are sharing the way you see fit. The most intimate thing you have is it.

This full-time commitment to math as a collaborative sport helps him deal with his anxiety. He's unsure about a lot of things. He wouldn't doubt it for months if he had a full proof in front of him. He is more at ease when his work is checked by others. Duminil-Copin said that he is not so self-assured.

He admits that he tends to over think. Everything, even when it comes to math. He oversaw the planning and construction of his new house, which he shares with his partner Séverine and their daughter. He wanted to modify even tiny details as it was built. He said that he was completely fixated. It would be sufficient to make the house half as beautiful as it is.

That is what proof-writing looks like for him, he makes improvements and gets obsessive about the details.

The advantages of obsessive attention to detail include: A view of the lake and its famous fountain can be seen from the family's new home built on a mountain. There is an important process in percolation theory that is reflected in the hexagonal tiles that cover the floor of his office. It was intentional.

He said that he likes that things are done in the correct way. I won't have to leave this house again. The language he uses mirrors how he describes his mathematical work.

Duminil-Copin is sitting on the patio and taking in his surroundings. He said he loved it. It's an invitation to think.

There are mountains around that have been there for millions of years. They will be there for a long time. It's reassuring to know that you're small.

He says that he needs to be in control despite his easy manner. He doesn't know how to build a new house or raise a child. He can't control the press coverage he gets for the Fields medal, it makes him uneasy. Things seem more manageable when you live in the mountains.

He uses math to heal. He can exercise more control because he knows what he is doing. The math he does is so pure that it is reassuring. He is able to forget everything else because of the intensity.

A Sharp Distinction

Much of what Duminil-Copin has proved has been known for decades because of experimental observations of percolation in real-life scenarios. Nobody knows how to prove that the answer is true.

Identifying the critical point at which the phase transition occurs is one part of Duminil-Copin's work.

Bernoulli percolation is a model in which the presence or absence of a black edge is determined by coin flips. That is not the case for most percolating models. The existence of a black edge in one part of the lattice will affect the other parts. Percolation models are more difficult to understand due to long-range dependency.

They are better models. Black edges can be used to represent communication channels between parts of a natural system. The Ising model is used to study the behavior of ferromagnets.

He had to understand the dependent percolation models before he was able to do so. He was able to identify the critical point for a large class of dependent percolation models. The answer to Bernoulli percolation can be found in the formula that he and the mathematician obtained. It was the beginning of the story, according to Duminil-Copin.

He wanted to know the behavior of the system above and below the critical point. The components on the lattice are all finite. How big are they? Duminil-Copin and Tassion showed that they are small. As the desired size of the island gets larger, the chances of finding a large connected component plummet. When you consider points that are farther away from each other, the probability that two meshes are in the same connected component decreases dramatically.

The finite connected components are also tiny.

Duminil-Copin and his colleagues used analysis and computer science to prove that sharpness is a property. He said that this result was one of the cornerstones of what he was aiming towards for a long time. The behavior of dependent percolation models above and below the critical point was assumed to be a hypothesis. Once Duminil-Copin had his proof, other results offered insight into certain quantities and features.

The precise moment of the phase transition is one of the most natural questions to ask. Tassion said that understanding that is the most difficult part for mathematicians. Most of the open problems are related to it.

The whole space might be invaded by the infinite connected component slowly. The percolation model is said to be constant at the critical point. When the phase transition occurs, the density might suddenly jump in a "discontinuous" way, with the infinite component permeating the whole system so that the fluid doesn't just find a path but can flow everywhere.

It's important to understand a given process. The transition from ice to water is not always straight forward. More energy needs to be added to the system for a while before the transformation happens. The discontinuity helps explain sudden changes in volume.

The transition for Bernoulli percolation is constant. Physical experiments and intuition show that dependent percolation models can be continuous or intermittent. Duminil-Copin wanted to get rid of that. The phase transition is always continuous in a certain regime for a class of dependent percolation models A year and a half later, he and another group of mathematicians proved that the same class of models has a phase transition.

Duminil-Copin has been able to apply his theory to other models like the Ising model. Half of the questions in the field were solved by Hugo.

Worked Out

His research is only a small part of what he does as a mathematician. He knows his duty for the community. He's a very responsible person.

He thinks about how his actions affect other people. Tassion said that he was extremely generous in collaborating. He has the ability to make you feel smart. The motivation is very high.

Writing a clear paper is a matter of respect for the mathematicians who will spend time reading it, and for those who might want to build on it or use it in their own work. Spending years refining a proof, distilling it to its essence, and cutting away unnecessary parts might be what that means. He simplified a proof in a way that would change how students learn about a key part of the field. Hairer said that he cares about explaining stuff and not just being the first to prove it.

Duminil-copin said you have a responsibility. Teaching, doing editorial work, and writing recommendation letters are all the same thing. Even if he's overwhelmed, activities that involve other people come first.

He feels as though he has been given another job, to act as a sort of ambassador, to properly show what research in math is like and why it is important, now that he has been awarded the Fields medal. He said you have no choice but to make this example.

He always likes to do math. He wants to show that dependent percolation models can satisfy certain symmetries at their critical point. If he and his colleagues can prove the existence of these symmetries, mathematicians will have more information to work with. You would get to know a lot about the model. It is the final step in 2D for me.

He and his co-authors took a big step towards showing conformal invariance. They might have gotten the breakthrough they needed earlier this year, according to Duminil-Copin. He said it wasn't clear if it would be strong enough to get a new result. He is looking forward to being able to think about it more deeply. The problem became very dear to me.