Mathematics is evolving: ‘There was this whole world of maths I hadn’t known about’

Climate change, cacophonies on social media, seismic shifts in society: sometimes the world feels overly complex and dynamic. What if you could zone in on just one feature of a system, and see how it evolves over time?
For Prof Laura DeMarco, such deep dives are the bread and butter of her research in dynamical systems at Harvard University in the United States.
“I work in a field of mathematics that is designed to describe anything that moves in time or evolves in time,” says DeMarco, who is a professor of mathematics and attached to the Radcliffe Institute.
DeMarco delivered the 2024 RIA Hamilton Lecture in Dublin recently, which celebrates the achievements of Irish mathematician William Rowan Hamilton, who made ground-breaking discoveries in algebra and dynamical systems in the 19th century. On October 16th, 1843, Hamilton etched his mathematical formula for quaternions at Broome Bridge by the Royal Canal in Cabra.
DeMarco uses maths to get to grips with the simple principles that govern complex systems, and to understand how predictable and stable those systems are likely to be. She offers the dance of planets in the solar system as a hypothetical example.
“Our solar system is complex, but how planets move in that system is governed by gravity and the rules of motion,” she says.
“We want to understand what is likely to happen in this system in the long term, like where will Earth be a few million years from now – is Earth’s orbit stable enough that it will be pretty much the same for millions of years until the sun dies or are there slight changes that could mean Earth will eventually crash into Mercury.”
Unwelcome celestial collisions aside, there is much beauty to be found in the principles that underscore complex systems, according to DeMarco, who also researches the self-repeating patterns called fractals, such as the Mandelbrot set.
“We have probably all seen pictures of these sets. They are very eye-catching, and they have been popular for maybe 40 years now.”
“The simple rules that create them result in beautifully complex images, but the really amazing part is that we still don’t fully understand it, and there are actually some really deep and difficult mathematical questions about these objects that we’re trying to understand.”
Today DeMarco is a member of the prestigious US National Academy of Sciences, and she has received the Alexanderson Award from the American Institute of Mathematics and the Satter Prize from the American Mathematical Society.
But back when she was a school student, maths wasn’t the only subject that caught her eye.
“I wasn’t super passionate about mathematics,” she says. “I was interested in a lot of different things, and math and science were two of them. But I also loved music and lots of other subjects too.”
Her “aha” moment in maths came at university, when one of her professors mentioned a research project they were working on. “I really had no idea that people did research in mathematics, that it was a thing,” she says.
“I think as students sometimes we have this impression that mathematics is done and you are learning these methods, these tools that were established hundreds of years ago, and now they’re very well understood. But when I dug into the textbooks, I could see maths was going in various different directions, and when I heard that my professor was doing research, I thought that sounded so exciting, there was this whole world of maths I hadn’t known about.”
A bit more digging and DeMarco was hooked, becoming a maths researcher herself. “There is so much to discover, so many interesting questions and it’s fascinating to explore them, to make those discoveries. The more research I did, the more I wanted to do.”
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DeMarco’s passion for maths and research fuels her energy and enthusiasm for lecturing students at Harvard and giving more public lectures too.
“I want people to know that maths isn’t scary – and that’s not done. We need people to research maths and answer these fascinating questions,” she says.
She is an advocate for women in maths, and she is encouraged to see women being more represented at more senior levels in academia, compared to when she was a student. “It has certainly evolved,” she says. “There are a lot more women now doing mathematics at this elite research level.”
DeMarco stresses that her current work on dynamical systems cannot currently answer questions about “messy” systemic challenges such as climate change, or the emergence of a disease, and she stands by the fundamental nature of the maths that she does.
“Ideally, some years down the road, what we discover about dynamical systems could be applied to understand transitions in systems with lots of variables, but for now the research in my field focuses on how simple elements evolve over time,” she says.
And while she sometimes harbours a little “applied envy” for the kind of mathematical research that focuses on solutions for near-term questions, DeMarco is a strong supporter of curiosity-inspired research that may – or may not – lead to new applications.
“We have to ask what do we value in society,” she says. “We value things like historical research, even though that may not have an immediate bearing on what we do today. But we want that research, we are interested in it, to understand the past. Through maths research, we can understand more about our world, our universe. Maybe it will provide a solution to some important question in the future. But even it doesn’t, isn’t it worth knowing?”
DeMarco’s advice for students with an interest in mathematics is to pursue it and see where it leads. “I think with an education in mathematics, you can do anything,” she says.
“To me one of the most powerful aspects of this subject: you can study mathematics to be a mathematician, it as a beautiful subject, but the skills that you learn by doing mathematics, the logical processing, the writing and communication and constructing proofs in ways that other people can understand, those are highly useful in other areas too.
“Of the students who study mathematics here at Harvard, some go on to be mathematicians, others are scooped up for jobs in other fields. Having done mathematics shows that they know how to work very hard, that they’ve learned to logically analyse things in a very careful way.”
One area where DeMarco sees mathematical skills being prized ever more is in biology, particularly as the field gets to grips with the complex and adaptive nature of biological systems, such as living organisms, wider ecosystems and diseases such as cancer.
“If someone studies mathematics and decides they want to apply it to medicine or biological research then they have this advantage of already having this core knowledge about maths and systems,” she says.
Given her love of it, it’s probably unsurprising that DeMarco racks her brain when asked what other path in life she might have taken had she not discovered the beauty of maths research.
“I think I would still be in fundamental research, maybe in chemistry or engineering,” she says. “But I’m happy to say that I love what I do, and honestly, my life is great.”
You can watch Prof Laura DeMarco’s Royal Irish Academy 2024 Hamilton Lecture on The Geometry and Algebra of the Mandelbrot Set at https://www.youtube.com/user/theroyalirishacademy