Number Theorist James Maynard Wins the Fields Medal

One of the best things that can happen to a mathematician is something that happens to James Maynard. He solved one of the discipline's oldest and most central problems after graduating from graduate school. He would have gained fame even beyond the cloistered world of pure math research if he'd achieved it. The only problem was that another mathematician had used a completely different method to prove the most headline- grabbing part of Maynard's result.

Maynard was immediately recognized by number theorists. It would not have been possible for a brand-new Ph.D. to solve it. Maynard knocked down one fundamental problem after another.

The 35-year-old mathematician was awarded the Fields Medal for his "spectacular contributions in analytic number theory." His work often leads to surprising breakthrough on important problems that seemed to be impossible with current techniques.

Maynard was surprised when he was informed of his Fields medal. I still think of myself as someone who is just starting out in mathematics. He draws inspiration from the work of previous Fields medal winners. It's amazing to be thrown up on this list with the mathematicians who inspired me as a kid.

Maynard cherishes the hope that his research life won't change too much, even though Fields medals shine a spotlight that usually follows a mathematician for the rest of their career. He said he was interested in the same problems and would continue to work on them.

The paper series about how prime numbers are distributed on the number line has been taking place recently. You could label four buckets with the digits 1, 3, 7 and 9 because all primes end in those digits. The buckets will all end up with the same number of primes, and this is true not only in base 10, but in any base as well. There are many questions about prime spacing that mathematicians don't know about how quickly the buckets start.

Maynard has shown that the buckets are able to leave faster than before. Kannan Soundararajan will give a lecture on Maynard's work at the award ceremony today.

Winning a Fields Medal is one of the highlights of mathematicians' lives. The most important event of the week has a lot of competition. Maynard and his partner, Eleanor Grant, are expecting a baby within a few days, which will allow Maynard to go to Helsinki quickly.

He said a lot of things are changing

On July 1, 2020, the following profile was published.

Simply Hard Questions

A health visitor came to James Maynard's house when he was three years old to check on his development. He was taken through a standard battery of tests during his visit. Maynard thought they weren't smart.

He explained why his solution was more interesting than hers when he was given a shape-sorting task. Gill Maynard wrote in an email that her son told her that the cow in his toy farm was sheep-sheep and that she liked it. He pulled out his Legos when he said the assessment was done.

Gill Maynard said in an interview that it was pretty memorable to see a three-year-old demolish a woman.

James didn't have discipline according to the assessor. She said that he would have problems in school if he went on like this.

Throughout Maynard's school years, there were many similar episodes. Maynard thought that his physics teacher was crazy when he used a rubric that only gave a third of the points for correct answers. Maynard wrote the answers and got them all correct and scored a third of a point. He thinks the teacher was fed up with him.

I was an annoying kid who would ask why. What's the reason? Maynard asked why all the time. He wanted to do his own thing or at least want justifications for things.

Maynard shrugged when he was warned off the problem he wanted to pursue, one of the most important questions about prime numbers.

The contrast between being simple and fundamental is appealing to me.

Andrew Granville, Maynard's mentor at the University of Montreal, said that he hoped Maynard wouldn't work on this full time.

Maynard sat down and said, "Let me try this idea and see where it takes me."

Ben Green of the University of Oxford said that the theorem Prompted a major reevaluation of how mathematicians think about the spacing between prime numbers

Maynard is drawn to questions about prime numbers that are easy for a high school student to understand but difficult for mathematicians to comprehend. The contrast between being simple and fundamental is appealing to him.

There are more questions now than before Maynard showed up. His first discovery about prime numbers and related structures was not a flash in the pan. Maynard is a leading number theorist.

He's had a steep upward trajectory in becoming a world-respected mathematician.

Maynard has slowed the progress of the author of a book on analytic number theory. About 150 additional pages have been added because of him.

Small Signals

Maynard was in Denver in January 2020 to receive the Frank Nelson Cole Prize in Number Theory. The prize committee cited three papers, all of which appeared in top mathematics journals, in Maynard's case.

Although he only had a day and a half for his Denver trip, there was a spring in his step. He smiled and said that he was still going on the thrill. It hasn't hit me yet. When he had to take a picture, he smiled happily. He said that people said he was unable to produce a smile in photos.

Maynard was going to wander the city and take pictures. He took up photography a few years ago to feel more connected to the many cities he travels to for work. He said that he went to Hong Kong in the summer and was hiking at dawn to take pictures.

Maynard passed through different phases of geology and astronomy as a child. He said that he is very bad at being interested in things. I have to either be obsessive about it or drop it completely.

Chris Maynard said his son tends not to stop until he reaches the limit of his ability. He is still not at that point in mathematics. I think that's what motivates him.

Maynard was always on the path that offered the most mathematics, even though his family was more interested in the humanities. He said it felt like the next step when he was at each stage.

His mathematical strength became evident in graduate school. His adviser, Roger Heath-Brown, said that their meetings felt more like collaboration than mentorship by the second half of his PhD studies. He said that he had never felt that way with a research student.

By the time Maynard left Oxford for a one-year post at the University of Montreal, he had begun to think about a way to understand the gaps between prime numbers. Primes get more scarce as you go along the number line. They behave like a collection of random numbers, so mathematicians expect them to be close to average. One of the most famous questions in mathematics is the twin primes conjecture, which states that there are infinitely many pairs of primes that differ by only two.

Maynard thought it might be possible to make progress on understanding prime gaps using a method described in a paper a decade earlier. Maynard thought it was possible to get more juice from the method. He said that he kept getting small signals that there was something there to be understood. I wanted to keep going until I could explain what I saw.

Maynard was discouraged by his adviser. I didn't think it was possible, what he was doing Even though I was very skeptical, James just laughed at it.

There was a big earthquake in the number theory world. The next best thing is that YitangZhang showed that there are infinitely many pairs of primes that are at least 70 million feet apart. The finding led to multiple job offers, invitations to lectures, news stories and even a documentary, as well as a new position on the faculty at the University of California, Santa Barbara.

James just laughed at it, even though I was very skeptical.

Maynard came up with a completely independent, more powerful approach to understanding prime gaps after six months of working on his own. Maynard applied his approach to triples, quadruples and larger collections. Soundararajan thought the result was too good to be true.

Maynard's euphoria was followed by a wave of fear that he had missed something. He said that he feels more productive when he is afraid that his result is wrong.

Maynard had to nail down every detail in his results. He told Maynard that nobody would believe him. No one can argue with you if you write this well.

Soundararajan said the result was a fantastic proof.

A young mathematician might be frightened by something that happened near the end of the process, when another mathematician came up with the same result. One of the most prolific and highly regarded mathematicians of the modern era is the University of California, Los Angeles's Terence Taoist. The problem caught the attention of Tao when he and other mathematicians collaborated to reduce the bound.

When he heard that a little-known 26 year-old had proved the same thing, he was very proud of his new result. The way he wrote it up, he had a better result than I did. He proved to be a bit stronger. Many mathematicians would assume that Tao had done the lion's share of the creative work, so he refrained from announcing his own work.

It is easy to imagine an alternative time period in whichZhang proved his result six months after Maynard instead of six months before. The glory would have gone to Maynard. Maynard doesn't feel bad about how things went. He said he was excited when he saw the result. Solving the problem is the main joy for me. I didn't think about it at all, 'Oh, if only I had done this differently.'

A Prime Game

Maynard doesn't wear his glasses when he walks from home to office. Sometimes the blurring helps him focus on mathematics, but sometimes it leads him to walk past Grant. Grant, a doctor in Oxford, said that there was a time when he ran up to a person who didn't look like him.

Maynard is in line with the stereotype of the absent minded professor. He wears an open collared white shirt and jeans almost every day. He said in an email that he was not the most fashion-orientated person. The mathematicians attending one of his talks showed up in Maynard uniforms.

He belies a lot of the stereotypes about mathematicians. He is affectionately called warm, fun-loving and outgoing by his coworkers. He used to bring his own coffee to work and brew it for the other number theorists. When he was a student at the Mathematical Sciences Research Institute in Berkeley, he shared a house with two other young mathematicians that was called the party house.

The new generation of number theorists are more social than their predecessors. The center of the group is him.

Number theorists applied Maynard's ideas to other problems after he proved his theorem. Maynard figured out how to address large prime gaps as well, improving upon estimates that had previously seen no significant progress. One of the cleverest tricks I have ever seen is Maynard's adaptation of his method to this new scenario.

Soundararajan said that anyone would be happy to have proved two of Maynard's ideas over the course of their career. He did it after finishing graduate school.

On this occasion he collaborated with Green and two other co-authors, he came up with roughly the same result at roughly the same time. Maynard and Tao have a tendency to come up with the same result. When he solved the number theory problem, he asked AndrewGranville if James would scoop him again.

I was very paranoid and asked if James hadn't snatched me again.

Maynard has given the number theory community plenty of proof that he is more than a clone of a famous mathematician. The Duffin-Schaeffer conjecture that asks which infinite collections of denominators produce fractions that do a good job of approximating irrational numbers was settled in the year 2019. It has been an approximation for a long time.

Maynard proved that there are infinitely many primes that don't have a 7. If you look at small numbers without any 7s, they are almost vanishingly rare, so showing that this sparse number set has infinitely many primes is not easy. People have been wondering about this for a long time, and no one was able to prove it.

Maynard came up with a proof for a very large base, which made sense in other bases. If you are in a base with a million different numerals instead of just 0 to 9, a restriction like "no 7s" has a smaller impact on the proof. Granville said Maynard's proof was "very elegant."

Maynard wanted to prove his theorem in base 10. He said that base 10 is the base that everyone talks about in daily life.

He reduced the base from 1,000,000 to 5000, then to 1000, then to 100. He said it became a game with himself about how sophisticated an argument could be. It was similar to betting machines or online games that give you a hit.

He was stuck on base 12 for a long time, worried that the final goal wouldn't work out. He got to base 10. He said that he was very happy to declare victory after dragging himself over the line.

Maynard had to come up with new ideas. His strength as a mathematician is shown in this picture.

A buzz of energy and anticipation has been created by this contribution and the others. There is no one else in analytic number theory that is generating more excitement.

He said that people were wondering what he would do next. Everything seems possible, that's what I think.