We met with Dr. Ian Stewart to discuss real-world applications for math, the joys of mentoring students, “things that go boing,” and much more. Enjoy!

Notable mathematician Dr. Ian Stewart explores the math behind “things that go boing” and how collaboration with math experts impacts manufacturing and other real-world applications. He discusses the math behind digital and biological networks, the joys of mentoring students, and how mathematicians appreciate each other’s work. Emeritus Professor of Mathematics at the University of Warwick, Dr. Stewart talks with Dr. Jed Macosko, academic director of AcademicInfluence.com and professor of physics at Wake Forest University.

For example, if I'm working with a maths graduate student or postdoc, they will have the background, but they might need some ideas about how to apply it, which bits to focus on, how to tackle the problem. If you're an experienced researcher in the subject, you have some tricks you can use. You've got some sort of feeling for the area which, early on, the graduate students may not have.” – Dr. Ian Stewart

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(**Editor’s Note:** The following transcript has been lightly edited to improve clarity.)

Jed Macosko: Hi. This is Jed Macosko at Wake Forest University and AcademicInfluence. And today we have, coming to us from England, Professor Ian Stewart, who’s a mathematician. And as a mathematician, I’m sure a lot of people wonder what is it you do and what usefulness is there in it? And I know you do some things, modeling, for example, animal gaits.

Do you collaborate with other labs?

I know, I was at UC Berkeley and Professor Bob Full does a lot of animal gaits and other people like him. Tell us a little bit about your collaborations with non-mathematicians.

Ian Stewart: Yeah, I’ve done quite a bit of collaborations with non-mathematicians. Probably the earliest one that was of any significance were actually... It was an industrial problem, working on springs, ordinary springs, things that go boing. They’re everywhere. And somebody has to make them and they’re made by... Certainly, over here, they’re made by small or medium-sized companies. They’re not huge manufacturers and they’re sandwiched between a big supplier of steel and the big companies that actually make the stuff they put their springs into, and so they’re kind of squeezed at both ends. So they had to make themselves very efficient. So I did some work with a whole group, there were 18 of us in the team eventually, and we came up with a new quality control method for spring... For the wire that they make the springs from.

Jed: Can you just tell us a little bit about that? That sounds fascinating. So what do you do?

Ian: We use chaos theory. There was an engineer in Sheffield, which is a city in England, slightly north of the middle, which is the home of many of these small companies. And he phoned me up one day, cold-calling out of the blue. He said, "I’ve just been reading one of your books and you describe the chaos theory method for analyzing data. And I tried it out on a problem we’ve had for years with springs and I think it solves the problem. Can we get together?"

So after about six months to-ing and fro-ing, he managed to secure some funding from the British government and we put together a team of wire manufacturers, spring manufacturers, engineers from Sheffield, two or three mathematicians, a control theory engineer and a man from the government who came along to make sure we were spending the money wisely. And for five years, we worked on this springs project and some related stuff. So that was fun.

In the middle, I collaborated with Jim Collins in the States who is now one of the leading lights in synthetic biology. He constructs man-made genetic circuits that are designed to do various tasks. But in those days, he was working on animal movement and human movement and we got together.

Recently, I have a long-term mathematical collaborator, Marty Golubitsky , in the States. We’ve been working together since 1983. About half of our research papers have both of our names on them, along with other people. So we have this long-term research relationship and we are currently in touch with four other different laboratories, a couple of physicists and various biological groups, and we’re looking at the mathematical structures in the big networks that biologists are now finding.

Everything in biology turns out to be a network. So the nervous system is a network of nerve cells. The way your genes regulate the development of your body when you were an embryo is a network of chemical reactions. One gene triggers another gene which triggers another one and so on. So they had been working on some of the experimental results. The biologists now have these wonderfully huge data sets on the entire nervous system of an organism. Okay, it’s a tiny, little worm, this thing called a nematode, but this has several thousand nerve cells in its connectome, as they call it, in its nervous system.

And so the mathematical challenge is, given the circuit diagram for the nervous system, how does it work? What does it do? And some work we’ve been doing, purely as a mathematical exercise on dynamics of networks, has turned out to be of some relevance to this question. So the group in New York, who first noticed this connection, have put together a consortium of five different research groups. We’re currently in this five-way collaboration with about 10 people.

But you can also get out there and talk to people, mix with people, get involved in other subjects. That's always quite an interesting learning curve…” – …

So math, nowadays, can be very collaborative. It doesn’t have to be. You can still sit on your own at your desk in your attic and think deep thoughts for years and years on end, and come up with wonderful theorems. But you can also get out there and talk to people, mix with people, get involved in other subjects. That’s always quite an interesting learning curve ’cause even when you have a lot in common, the biologists use different language from the mathematicians even when they’re talking about the same thing, so it takes a while to get comfortable.

Jed: Well, you seem to have gotten comfortable with nematodes and gene regulation.

Ian: Oh, yeah. Well, I’m a bit of the butterfly brain.

Jed: Oh, that’s good.

Ian: Yeah. I like doing lots of different things.

Jed: Tell us, just in layman’s terms, how you solved the long-lasting problem in the spring industry? What about chaos theory allowed you to solve a problem? What was the problem? Was it that they were breaking and nobody could quality control that?

Ian: Okay. No, the problem was subtle. The problem was if you... Basically, you make a spring by passing a wire through a coiling machine at high speed and the coiling machine just bends the wire a little bit, and then the wire spins off in a helical coil and it’s all feeding through at 10 springs a second, so high speed. And if the wire is not good, it makes very weird-shaped springs. They’re just useless. But they couldn’t find a test for whether it was good or not, except trying to coil springs with it. And when you try it and it doesn’t work, then your very highly paid, experienced coiling machine operator tries again and adjusts things a bit, gets his spanner out and changes things and tries again and it still doesn’t work and after two days of this, you’ve wasted a lot of time, a lot of money, and you still haven’t got any springs.

There were quality control methods, but they were largely about the tensile strength, make sure it won’t break, things like that, chemical properties of the wire, the metal, and they didn’t solve the problem. They didn’t seem able to... They could tell the difference between really bad and really good, but they couldn’t address the interesting bit in the middle.

Now, what the engineers in Sheffield had realized was they could get part way by forcing wire to make a helical coil, whether it wanted to or not, by coiling it along a rod, a metal rod. You have a rotating rod and the wire moves along and it gets wound. It’s like winding spaghetti on your fork, but done with wire on a rod.

And what they found was if it was good wire, then the spacing of the coils was very, very precise and consistent. And if it was this middling quality that was difficult, some spaces were bigger than others, some spaces were smaller than others, so somehow the variation in the strength of the wire was causing variations in the spacing of the coils and if that got too bad, you couldn’t make good springs.

But when they tried to characterize the variability of these spacings using standard statistical methods, it still didn’t work. You couldn’t tell the difference between something that would coil and something that wouldn’t. Then they realized there’s some information they were missing. It’s not just the list of spacings shuffled up, which is what statistics looks at, it’s the sequence of spacings along the coil.

Jed: And that’s where chaos theory comes in.

Ian: And that’s where the chaos theory comes in.

Jed: Okay. Very cool.

Ian: That’s how we do it.

Jed: Wow. And it took five years to figure that out. [chuckle]

Ian: Yeah, yeah. Well, Len Reynolds, the engineer who phoned me up, kind of had this picture in his head from day one, but you have to do the leg work. You have to actually do the experiments, do the math, try it out, get lots of samples of wire, run them through the test, put them on a coiling machine, report to the government every three months telling them what you’re doing, all of that stuff. And yeah, it took a while. [chuckle]

Jed: Now, switching gears, when you collaborate with somebody who’s doing biological systems, the nematodes, connectome, or probably an easier example to understand, the walking gait of a horse or something, do you also collaborate with graduate students who are doing the measurements on the horses or whatnot?

Ian: Oh, yes.

Jed: And why can’t they just do the math on their own? Do they just not have enough mathematical background, being biologists in training or something?

Ian: Okay, it depends on... For example, if I’m working with a maths graduate student or postdoc, they will have the background, but they might need some ideas about how to apply it, which bits to focus on, how to tackle the problem. If you’re an experienced researcher in the subject, you have some tricks you can use. You’ve got some sort of feeling for the area which, early on, the graduate students may not have. Now, my experience with graduate... I’ve had about 30 PhD students in my career. [chuckle] A good many of them, at least 10, were better mathematicians than I am and I say this... Mathematicians kind of know where in the pecking order they sit. These are very brilliant students who have gone on to mathematical careers in universities but early on, they need someone to bounce ideas off. But what I found with the really good ones, after a while, instead of me telling them what to do and they go away and do it, they would come to me and tell me what they’d done.

[laughter]

Okay, I’ve been working on the problem. This is what I’ve done. I had this idea, so I tried this out. It seems to work, but I can’t quite sort out what’s going on here. Something’s not right." And then, of course, you turn on your experienced researcher switch and you start asking questions and making suggestions, half of which you don’t actually quite understand in detail what the problem is, but it gets them thinking about it.

So a very bright graduate... I mean, anyone who gets to graduate definitely is pretty bright. [chuckle] They have to be. Even bright undergraduates. These are very exciting people to work with because they haven’t been doing it for the last 25 years or, in my case, 50 years. They’re new, they’re enthusiastic. They’re very bright.

In fact, Stephen Smale , one of the world’s great mathematicians, said early on in his career, he solved a huge difficult frontier problem in topology, and somebody asked him how did he do this, and he said, "Well, I was so new to the subject, I was too stupid to realize it was hard. So I wasn’t put off by what everybody else knew, that this is really difficult. So I had an idea that I thought might work and I tried it, and it seemed to make progress and solved the problem." So constructive ignorance can actually be useful.

Jed: Yes, I really admire you mathematicians. And there are a lot of really great mathematicians out there that... I don’t know if you saw the list at AcademicInfluence, but one of the ones I noticed on there was Terry Tao. You probably of course are familiar with his...

Ian: I know Terry... Oh, yeah, I know, he’s amazing. He is absolutely extraordinary.

Jed: What were some of the things he was known for? ’Cause I knew him back when he was 15 years old. We were in a program together in the summer, but I never figured out what it was he was famous for.

Ian: Well, he works on a lot of different things, but one of the things that kind of links together a lot of what he’s interested in, not all of it, but a lot, is Fourier analysis, which engineers certainly will have heard of. So this is how you take some sort of signal that’s varying in time, and represent it as a combination of pure tone sine curves. So nice regular, wiggly curves with different frequencies and amplitudes. And you add all those up and you get a very complicated signal. So this is absolutely... It goes right back to Joseph Fourier in France in the 1800s, working on heat, oddly enough, but it’s used for vibrations for earthquake signals, for if you want to find out where to prospect for oil. You set off explosives, and the waves come back and you Fourier-analyze those, and do related things. And then there are...

Anyway, so Terry Tao kind of applies Fourier methods all over the place in mathematics, and in particular to number theory. So he has some very interesting results with Ben Green on prime numbers and questions about distribution of prime numbers, which he approaches through this Fourier analysis method, which you wouldn’t think those two areas come together.

Of course, one of the messages from mathematics nowadays is any two areas can come together. It’s a very highly interconnected... Maybe there’s a third one in between that links them, but all areas of mathematics have connections with all other areas, and at any moment... I work in dynamical systems. **I think I’m working on some problem about differential equations and, whoops, no, suddenly I’m working on some algebra problem or some graph theory problem, which happens to be the key to that differential equations structure. And so this is what makes it fun. This is what it’s all about.**

Jed: Well, it is just amazing hearing you as a mathematician discussing your field and the fun you have in it, and the usefulness it has when you collaborate with other people, when you work on problems, but also just the beauty of it, just for the sheer sake of the beauty. So we really appreciate you coming on this show, Professor Stewart. Thank you for taking some time with us today.

Ian: Thank you, Jed.

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