To infinity and beyond

Professor of Mathematics Chris Good first became aware of infinity on a beach in Brittany when he was about five. 

‘I was with my brother, playing in the sea, and my dad went to fetch us ice creams,’ he recalls. ‘I came out of the water and dried my hands on the sand. But, of course, that made them sandy, so I went and washed them in the sea. Then I dried them again on the sand, which made them sandy again, so I had to go back down to the water. I remember thinking, “this will carry on forever”. It was my first view of infinity.’ 

The memory also provided the first joke for his Inaugural Lecture, which he delivered to a packed house earlier this year. ‘What I said was that it was clear back then that I was going to be a pure mathematician – because an applied mathematician would have used a towel!’ 

Professor Good had been unsure about including jokes in his address, so was relieved when this elicited a hearty laugh from the audience. As well as amusing asides, he also talked about the likes of rugby, bread-making and family as he charted his journey from that five-year-old boy to the acclaimed academic he is today. 

His address was part of a series of Inaugural Lectures run by the College of Engineering and Physical Sciences to showcase its leading scientists who are pushing the boundaries in their disciplines. These events, which are open to the public and free of charge, mark an academic’s promotion to Professor, but also provide a rare opportunity to hear first-hand about their life and research. 

Chris’s introduction to mathematics at school – times tables – didn’t auger particularly well. In tests, he rarely achieved full marks. ‘I’m no good at mental arithmetic even now – yet I knew that that’s not really what mathematics was about.’

By the time he was 13, his teacher was predicting Chris would read mathematics at Oxford – which is just what he went on to do. He studied for his PhD there, too. 

‘It was also at school that I decided I wanted to do pure mathematics, rather than applied or statistics,’ he says. 

Today, Chris has more than 50 research articles to his name in set-theoretic topology and topological dynamics, yet he very nearly gave up research almost before he’d begun. 

‘I was at Oxford and I’d spent about eight months working really hard to solve a particular problem that I was interested in,’ he remembers. ‘I had a good idea how to solve it and got to the point of writing a proof. When it was finished I went for a walk, and halfway round Oxford I realised there was a subtle mistake in it! I carried on working on the problem, but then I discovered someone else had solved it using the same idea I’d had. I was close to giving up research then.’ 

But he ‘clawed’ his way back, and since joining the School of Mathematics at Birmingham in 1995 he has proved many theorems, both on his own and in collaboration with colleagues around the world. 

Chris started his research career in topology, a branch of mathematics sometimes called ‘rubber sheet geometry,’ in which two objects are considered equivalent if they can be continuously deformed into one another through such motions in space as bending, twisting, stretching, and shrinking. The main topics of interest in topology are the properties that remain unchanged by such continuous deformations. 

‘Mathematicians like functions, because they tell you, given your inputs, what your outputs are going to be. For example, if you invest money at a certain interest rate, you know how much money you’ll get out at the end. So, it’s a way of describing the world and talking about the world in a quantitative manner. 

‘Normally we talk about function, but don’t talk about where the input values come from, but it turns out that the structure of what the input values look like can affect the behaviour of the function. A topologist thinks about the structure of the input values as much as the function.’ 

During his career, Chris’s research focus has shifted from general topology to topological dynamics, looking at the likes of chaos theory and population dynamics. His most recent work, which he cited in his Inaugural Lecture, is the notion of shadowing in a dynamical system. 

‘Let’s assume your function, the investment of money, is deterministic on, say, mortgage and interest rates, and you track it. Then you take the money out and reinvest it and track it again. To do this long-term, you would put the information into a computer, which should spit out the exact answer at the end. However, there are all sorts of rounding-to-the-nearest-decimal point errors when you use a computer, so you get a tiny little jump, or error, every time. This results in what we call a pseudo-orbit, rather than an exact or real orbit. In some systems – those with shadowing – it turns out that there is always a real orbit that’s close to the pseudo-orbit, so the approximate behaviour you’re seeing on the computer screen really does reflect real behaviour.’ 

Although ‘shadowing’ is a well-established phenomenon, Chris and one of his postdoctoral fellows discovered a complete characterisation of systems that have this. 

‘We explained that all these systems that have this can be built up from the same basic building blocks that are a fundamental type. And that’s something that’s new.’ 

Along with his research, Chris – who is Deputy Head of School – is also passionate about teaching: he was award the first Head of School’s Excellence in Teaching Award. ‘The quid pro quo for being able to conduct esoteric research is teaching, but I really enjoy it. I also enjoy the admin aspect of my role, because it’s about problem solving – just like mathematics.’ 

Would he one day like to get involved in research that is less ‘esoteric’ and more applied? 

‘I used to be quite sniffy and say that I did what I did because it was fun,’ says Chris, ‘but now I would love to do something that had some application, such as topological data analysis relating to cancer studies. That might come off one day, and if it did it would be a blast.’