When James Maynard was a child, he didn’t feel a singular, driving force toward studying mathematics, even though he liked the subject.

Of course, he had his phases: Dinosaurs. Tractors. LEGO. Maynard, who was one of four recipients of the 2022 Fields Medal — a top international award for mathematicians under the age of 40 — says he’s always “enjoyed what I enjoy and didn’t enjoy what I didn’t enjoy, and I just focus on those things that I like.”

As an adult, that means mathematics.

Maynard, a research professor at the University of Oxford, is the 2025 recipient of the Rev. Joseph Carrier, C.S.C., Science Medal at the University of Notre Dame, an award given in recognition of sustained, outstanding achievements in the sciences. He will accept the medal and give a public presentation in Jordan Hall on Monday, Sept. 15.

His primary area of interest within mathematics is in a field called number theory. At its most basic, number theory is the study of whole numbers and the relationships between them. It explores patterns, like how prime numbers are distributed, or how numbers can be divided evenly.

Let’s go back to fourth or fifth grade: Prime numbers are those greater than 1 that cannot be divided by any whole number other than itself and 1. They are fundamental objects in mathematics, and though they’re introduced to young children, they are also completely mysterious. Full understanding of prime numbers has resisted the efforts of mathematicians studying them for thousands of years.

In March, Maynard shared the significance of understanding prime numbers, as well as the history of the field, during the inaugural public talk of the Pittsburgh Mathematical Horizons Lecture Series, sponsored by the Benter Foundation at the University of Pittsburgh.

“Why care about primes?” he asked, dressed in a white oxford and khaki pants — a variation of what the mathematics community jokes is his usual “uniform”: a white button-down oxford and jeans. “This may seem like some weird, geeky thing to study.”

Though it may seem abstract, understanding prime numbers plays a key role in computer security and encryption, as well as within digital communications. The key reason is that primes are easy to multiply together, but difficult to pull apart. Encryption works by multiplying prime numbers, which is easy for a computer to do. However, to break the encryption, a hacker (or a hacker’s computer) would need to figure out the two original prime numbers using factoring, which is difficult even for mainframe computers.

Maynard’s first major breakthrough, and a key part of why he was awarded the Fields Medal, addressed a problem that had intrigued mathematicians for centuries: How close together can prime numbers come to each other? This touches on mathematics’ famous twin prime conjecture, which posits that there are infinitely many pairs of primes that differ by exactly two—like 11 and 13.

Working on solving this question was almost an obsession, or at minimum, a quiet compulsion for him. When he’s doing research, he says he needs to get back to that level of obsession he had with his favorite subjects as a child (“if it was that I was very into tractors, then I would live and breathe tractors,” he said). He lived and breathed the twin prime conjecture proof, physically immersing himself in the problem using both pen and paper and computers, and stacking scraps of paper on top of notes in his office, where he tries to eliminate all other distractions. He brings his smartphone into the room with him, but once he’s focused on math problems, it becomes irrelevant.

“It’s the combination of feeling that these are really inherently important fundamental objects, but still we don’t understand them, that’s mind-blowing for me,” he said. “We know as you go further and further down the number line, the primes get rarer and rarer.

“But we believe, and can now prove, that occasionally you get primes which come unusually close together.”

Though some questions remain unanswered, Maynard proved in November 2013 that there are infinitely many pairs of prime numbers that are at most 600 numbers apart.

“From the mathematician’s point of view, it was the new method that was the exciting thing,” Maynard said.

As he pointed out, 600 still sounds like a wide gap, but it’s really not: Mathematician Yitang Zhang, who earned his bachelor’s and master’s degrees from Peking University and his doctorate from Purdue University, had shown in April 2013 that some prime numbers are at most 70 million units apart. That result was the first time any mathematician had proven that there was any fixed gap between prime numbers that occurred an infinite number of times.

Remarkably, Maynard didn’t know that Zhang had been working on the same problem at about the same time, using different methods. Zhang used an advanced filtering technique, called a sieve, and deep number theoretical tools to arrive at his solution. It was a powerful solution, but complex and not easy to adapt to other situations.

Maynard’s result seven months later used a new type of sieve that was much easier to use. He essentially invented a new type of filter that didn’t just search for pairs of primes, but also for triplets, quadruplets, or even more.

Though some in the field wondered if Maynard was competing with Zhang, he firmly rejected the idea of competition.

“People like to talk about competition in mathematics, but I don’t think of it that way,” he said. “We’re all trying to understand the same things. It’s more like a shared journey than a race.”

It’s a journey that those in the field enjoy, and what keeps them searching for answers. Still, Maynard stands out, said Roger Heath-Brown, emeritus professor of pure mathematics at the Mathematical Institute at Oxford University. Any good mathematician should be a master of a good range of techniques; they may need to be highly competent computationally. And they have to put a lot of hard work into their research, Heath-Brown said.

“James scores exceptionally highly on all these counts. In addition, he thinks deeply and clearly about his mathematics. That clarity of thought is evident from his writing, where technical details appear to be handled with ease,” he said. “But most of all, it is his ability to bring radical new ideas to his mathematics that makes him a star.”

The biggest question in math: Proving the Riemann Hypothesis

Maynard has more recently turned his attention to the Riemann Hypothesis, which lies at the heart of prime number theory. It’s based on a mathematical object called the Riemann zeta function, which is similar to a complicated machine that outputs numbers based on inputs, and functions like a map that shows a mathematician where prime numbers are hiding.

The 165-year-old hypothesis says that all of the of the interesting “zero points” of the zeta function, or where the function equals zero, lie along a perfectly straight vertical line, called the critical line, in a complex number system. The Riemann zeta function has been rigorously studied and many of its properties proven; the Riemann Hypothesis, specifically regarding where the function equals zero, has not. But mathematicians agree that it can be proven.

“This is the most important problem in the whole of mathematics,” Maynard said. “We don’t quite have the right inroads to think about it, so we have to accept somewhat weak results in our attempts to try to understand it.”

The solution for the Riemann Hypothesis has high stakes: The Clay Mathematics Institute will reward the first mathematician (or group of mathematicians) $1 million to solve it. Maynard and Larry Guth, the Claude E. Shannon Professor of Mathematics at the Massachusetts Institute of Technology, published a proof in 2024 that provides better estimates of how many prime numbers exist on short intervals in a small stretch of numbers, even far out on the number line.

They also determined a better estimate of how many “bad” zeros the zeta function might have off the critical line — where they shouldn’t be if the Riemann Hypothesis is true. Their solution provides better ways to control when certain number theory formulas are unusually large, and helps mathematicians better understand the behavior of prime numbers and the zeta function.

Despite making some headway, Maynard compares proving the hypothesis to a “massive mountain” for which mathematicians don’t have the tools to climb.

“So we have to content ourselves with finding paths around the mountain,” he said.

Mathematics: Path to stardom?

Maynard may do most of his work in a silent office, but that doesn’t mean he’s a brilliant loner. He’s friendly, willing to share knowledge, and self-effacing. These qualities make him easily approachable.

“In addition to being brilliant and extremely creative, he is a very generous person and very open to collaboration,” said Kevin Ford, professor of mathematics at the University of Illinois Urbana-Champaign. “He is an elite mathematician who is not elitist, and has shared many of his ideas with students and post-doctoral workers.

And so people recognize him, and this is something Maynard says he’s been particularly surprised about. One time a mathematics fan recognized Maynard when he was at dinner with his parents. While Maynard was visiting the University of Pittsburgh campus, Isabella Canals, a senior majoring in economics and mathematics, noticed him outside and stopped to ask if he was “that mathematician” giving an upcoming lecture, which she did attend.

“It’s still very weird for me,” he said. “I have to pinch myself sometimes when people stop me on the street to ask for a photo.

“I still find it a bit surprising, weird and like lots of fun, but definitely, completely not what I was expecting when I imagined my career as a mathematician: having interviews, having people stop for photos, suddenly being this person that people want to listen to in some ways.”

Though Maynard admits that receiving accolades like the Fields Medal, or being chosen to present lectures and other awards, like Notre Dame’s Carrier Medal, feels amazing, he truly does not think about awards while doing his research. It’s about mathematics, about teasing out the possibilities of what he considers a fascinating problem.

The prizes have made him recognizable, but thinking about them is not in the spirit of the award.

“The worst way to prove a great theorem is to wake up and say, okay, today I’m going to prove a great theorem,” he said.

But prizes are significant for making the public aware of the importance of science, Maynard said. He thinks back to his childhood and understanding that major prizes, like the Nobel prizes or the Fields Medal, demonstrate the type of work that is highly important, and worth funding and supporting. High-profile prizes have been quite successful at putting serious research problems in the consciousness of the public, he said.

“I think awards ... can be these really clear signifiers of, ‘here's some academia coming together and saying, here's some really great results,’” he said. “It's a very strong signal to young people and to the wider public, that this is what real science is.”

It’s not always about mathematics

Though Maynard enjoys spending his time chipping away at some of his field’s most pressing questions, he likes to balance work with family and fun.

Among his favorite hobbies?

“I again have sort of geeky obsessions: photography and coffee,” he said and chuckled. His photography is more about process than product, and he tends to keep the finished product to himself, even as he enjoys finding creative angles for his photographs.

“I try and come up with sort of artistic ways of focusing on small details or unusual perspectives, rather than just the sort of standard photo of the big cathedral,” he said.

Modern art is another passion, and he’s drawn to abstract impressionism, particularly by Jackson Pollack, and surrealism from the middle of the 20th century, starting with Salvadore Dali.

Outside the world of prime numbers and proofs, Maynard keeps up with his two children, ages 3 and 1, with his partner Eleanor Grant, a physician.

They are his focus in his “off” time, and his family sometimes travels with him. While in Pittsburgh, for instance, he visited the Carnegie Museums of Natural History, where his son enjoyed peering at the dinosaur skeletons.

Accolades aside, Maynard is grounded by his family life, his colleagues shared.

“When he received his Fields Medal, he remarked to me that it was a really important event in his life, but it was only the second most important event taking place that week; two days later his first child was born,” Ford said.

Originally published by Deanna Csomo Ferrell at science.nd.edu on August 12, 2025.