It’s a warm summer afternoon in early July and we’re sat in our respective homes, on our laptops, waiting to talk to Martin. We’re both feeling a little nervous as Martin is renowned for being one of the top researchers in the field of stochastic analysis, maybe in the maths community full stop. The notification pops up on screen, Martin has arrived. He greets us with a friendly hello, and instantly puts us at ease with his polite and cheerful demeanour. Martin is speaking to us remotely from a hotel garden in Helsinki; he’s in Finland for the International Congress of Mathematicians (ICM) where this year’s Fields medallists will shortly be announced.
The ICM takes place every 4 years, and is where the Fields, Abacus, Gauss and Chern medals are awarded. This year the congress was due to take place in St Petersburg, but a decision to boycott Russia over the invasion of Ukraine led to a series of satellite conferences being hosted in neighbouring countries. Martin has acted as satellite coordinator and chair of the programme committee for this year’s congress. His role involved inviting plenary speakers, and helping to transition the conference to a hybrid format. But this isn’t Martin’s first involvement with the ICM—he won the Fields medal at the 2014 congress for his work on a regularity theory for stochastic partial differential equations.
Photo: King Ming Lam
The long and winding road
The Fields medal is probably the most famous award on Martin’s long list of achievements, but Martin’s first taste of success was developing Amadeus, a popular sound editing software package, which can be used for creating podcasts or original musical scores. He started developing the software as a teenager, and entered it into a competition that he won. At the time it seemed like the perfect outlet to practise his newfound coding skills and indulge his love of music: “I like classical pop rock music basically, you know, like Pink Floyd or Dire Straits or the Beatles”.
Amadeus continues to be something Martin is well known for. It’s hard to understand how Martin has time for all of this: developing software is a full time job in itself; doing it alongside being a prizewinning mathematician is extraordinary. Martin is frank: “I haven’t had much time the last couple of years,” he says, explaining that lately all he has time for is to maintain his software, but “zero time for development”.
Given his early success developing software, was he ever tempted to study computer science at university?
“Yeah, I was actually. I guess the reason why I didn’t study computer science as an undergrad was that I kind of arrogantly believed I already knew it somehow,” Martin says with a sheepish laugh.
Ernst Hairer, professor of mathematics at the University of Geneva. Photo: Renate Schmid, CC BY-SA 2.0 DE
Martin instead decided to study physics as an undergraduate, and went on to do his PhD in the physics department as well. For Martin this was something of a deliberate teenage rebellion—Martin’s father is a distinguished maths professor with his own list of accolades to his name. Perhaps it wasn’t the most potent of rebellions—Martin admits that during his undergraduate he took many classes from the maths department. Further, he says his PhD would’ve been taken under the maths department at most universities.
Even after his PhD, software development was always in the back of his mind. “It was always a plan B. After my PhD, when I decided to continue in academia, I sort of knew it was tough getting permanent jobs in academia. So the plan B was, well, if it doesn’t work out, then I’ll just do software development.”
It’s interesting to hear an accoladed academic talking about job precarity, and reassuring too: this is a concern that many doctoral students contend with, and a reality many postdoctoral researchers have to face. Despite these concerns, Martin did continue in academia, finally moving to the maths department for his postdoc.
Martin really enjoys working in academia. For him it provides one of the best things about being a mathematician: “the freedom to do basically what you want, at least to some extent.” Martin continues: “In some sense you don’t really have a boss as an academic. I mean, officially, inside the university structures you’d always have a head of department or somebody, but they don’t tell you what to do. So you can do what you want to a large extent, and I think that freedom is really quite precious.”
Spoilers for Martin’s next paper. Photo: King Ming Lam
Every little thing
For a man whose list of awards is staggeringly long, Martin is generous with his time and easy to talk to, happy to chat about everything we throw at him. The only thing he seems a little shy about talking about is the awards themselves. We asked him what his biggest achievement is, or what prize he is most proud of, expecting him to mention the Fields medal, or perhaps the Breakthrough Prize which he won in 2021. He replies that nothing could top figuring out his regularity theory.
This is something that comes up a few times while we speak to him—the joy of figuring out a problem. Where does that joy come from? Is it the surety in knowing that a theory is finished, that the wrinkles have been smoothed out, or even that the work is ready to be published?
For Martin it’s the moment right at the beginning, when it all starts to click together. “You sort of have a few days, where you’re scribbling around like mad and trying to convince yourself that this actually has a chance of working. I convinced myself relatively quickly that it would work and be a big deal somehow. Obviously it takes a lot of time to work things out, but I knew that it would work out, so it’s not so much putting in the last piece.”
Martin’s work focuses on understanding small scale randomness through large scale behaviour of systems, in particular stochastic partial differential equations. A stochastic partial differential equation (SPDE for those in the know) is a partial differential equation where some of the parameters are random variables, or where the solution can be written in terms of a random variable.
One of the most famous SPDEs Martin works with is the Kardar–Parisi–Zhang (KPZ) equation. Imagine setting a piece of paper on fire, from the bottom edge. The top edge of the burnt part will move up in a way that’s mostly predictable if you’re standing far away, but will look random and jagged if you’re standing very close. The KPZ equation describes situations like this: if we write $h(x,t)$ for the function describing the height of the burnt part at time $t$, it is governed by the equation
\[
\frac{\partial h}{\partial t} = \nu \frac{\partial^2 h}{\partial x^2} + \frac{\lambda}{2} \left(\frac{\partial h}{\partial x}\right)^2 + \eta (x,t),
\]
where $\lambda$ and $\nu$ are constants representing how inflammable the paper is. Importantly here, $\eta(x,t)$ is a random ‘white noise’ term. If we leave it out, we’re left with just a partial differential equation, or PDE. These are tricky, but solvable, and we can work out all sorts of nice things about the solutions. For example, this PDE would have a solution whose graph is smooth. However, including the white noise throws a real spanner in the works, and for decades mathematicians were mostly stumped about how to analyse the behaviour of $h$. Martin found a way to make the equations make sense, which comes down to ‘subtracting infinity from infinity’.
Martin’s regularity theory for SPDEs astounded the maths community—his ideas seemed to come out of the blue. One of the reasons this theory surprised the community is that his breakthrough idea came from physics—using finite series of wavelets (more typically used to encode information in digital files) to understand the behaviour of SPDEs .
Martin’s work continues to focus on stochastic processes. One of his current research projects is exploring discrete systems which are updated according to simple rules. “You try to understand the global large-scale behaviour of these things, and the limits you can get in that way. In some restricted context, people have a pretty good idea of what is happening, and then there are cases which are completely open. There are quite a lot of things there still—it’s one of the big areas of probability theory. The first result of this type would have been the central limit theorem… which goes way back to the 18th century.” Describing how trends in maths evolve, Martin continues: “People have always been interested in these sorts of areas. You have periods of stagnation, and periods where people come up with a technique and a flurry of activity. That’s kind of the way mathematics works.”
Distinguish yourself by wearing colourful socks. Photo: King Ming Lam
In my life
For the last five years Martin has been a professor at Imperial College London where he is a researcher, PhD supervisor, and occasional lecturer of a masters course on his breakthrough work. Martin regularly delivers summer schools, which are lecture courses typically attended by PhD students and early career researchers. For Martin, teaching summer schools is a really important part of being a researcher, as sharing knowledge is fundamental to advancing mathematical ideas. “If you’re just doing something alone in a corner and don’t explain it to anyone, there’s not much point, right?”
Two years of the pandemic have changed how mathematics is taught, developed and discussed. After all that time Martin is excited to be back attending events in person. “In the beginning there was a lot of enthusiasm for online seminars and workshops. It works pretty well. You can even give a blackboard talk using your iPad.” But, he adds, “it gets old pretty quickly.”
The pretty part of Imperial College London
A new hybrid method of working does have its advantages: it makes it easier to collaborate with fellow researchers overseas, to attend conferences in faraway places and it can be more accessible for people who might struggle to come into the office, whether due to childcare responsibilities or health reasons. However for many of us the joy of doing maths is doing it with others, and it’s difficult to mimic the fluid exchange of ideas that occurs in person. “Just catching up with people, to figure out what’s actually going on in the community: you can’t quite figure it out by only listening to talks.”
How does Martin think the pandemic years will affect the future of how mathematics is done? “We seem to be converging on a model,” Martin says, “where it’s an option for the speaker to deliver it remotely or come in in person.” This way of working certainly seems to offer a good compromise of social interaction and accessibility.
Martin with fellow 2014 Fields medallists Artur Avila, Maryam Mirzakhani and Manjul Bhargava (and Maryam’s daughter Anahita, who is yet to win a Fields medal). Photo: Wikimedia Commons user Monsoon0, CC BY-SA 4.0
Come together
When Hairer won the Fields medal in 2014, he won alongside Artur Avila (interviewed in Chalkdust issue 02), Manjul Bhargava and Maryam Mirzakhani. Avila was the first South American to win the award, Bhargava the first person of Indian origin and Mirzakhani the first woman, all in the 78th year since the medal’s inception. Since we spoke, this year’s winners have been announced as Hugo Duminil-Copin, June Huh, James Maynard and Maryna Viazovska—the second woman to win the award.
Stem subjects, and in particular maths, have a reputation for struggling with diversity. Does Martin think the ICM has a responsibility to do more to increase gender and minority representation in the sciences? Martin is thoughtful, “it’s tricky,” he says, especially as it goes beyond gender, with under-representation based on ethnicity, socioeconomic and cultural backgrounds too. “There’s a leaky pipeline, the proportion of women gets smaller over time. It’s difficult to find a solution; the one thing that you definitely don’t want to do for something like a Fields medal is to put some kind of quota, because that just devalues it.”
“So the right way to do it is as things percolate up; naturally there will be more diversity.” Martin is hopeful that gender representation is improving in maths. “You see that already in the ICM—if you look at the proportion of female speakers at this ICM it’s about 25%. At the last ICM it was like 15%.”
Waiting for things to percolate up only works if we have the structures in place to support under-represented groups once they reach each stage of the pipeline. This is a difficult task—one important thing is to encourage everyone from a young age that they can do maths, and that it is a viable vocation regardless of background.
Looking back we wonder what advice Martin would have for his 18-year-old self, just embarking on his mathematical journey?
“Maybe I should have paid a bit more attention in the algebra classes.” More seriously, Martin continues with advice to anyone just starting their undergraduate degree: “follow your nose, and do the things which you find interesting.”
Photo: King Ming Lam
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