Dr Michael Coons
The University of Newcastle
Dr Michael Coons takes us through the things that pique his mathematical interest and the best piece of advice he can give to researchers of the future
Can you tell us about your work? What drives your interest in this field?
My research interests lie in pure mathematics as broadly defined. At heart I am a problem solver, and due to this, my research has connections to number theory, combinatorics, theoretical computer science, analysis, algebra, and dynamical systems. I have special interest and expertise in Diophantine approximation results related to structures in theoretical computer science and dynamics such as finite automata, regular sequences and block substitutions. My interest is piqued when a problem has many facets that are of interest to several areas within mathematics.
I often work on several problems simultaneously, both as a sole-researcher and in collaboration. My current research is mostly focussed on growth aspects of regular sequences—generalisations of automatic sequences. I am interested in how the maximal values of regular sequences in certain intervals can shed light on the finiteness conjecture for finite sets of integer matrices, and if the theory of Mahler functions can be used to give answers in this area.
What are the most interesting “big questions” or challenges facing researchers in your area?
Pure mathematics is full of questions—as is number theory. For my part, the problems I find most important are questions at the interface of theoretical computer science and number theory. The biggest of these is known as Borel’s conjecture. It states that the base expansion of a real irrational algebraic number is normal, meaning that it is essentially random. The point here is to examine how people think about things (where the easy things are algebraic using the standard algebraic operations like ‘plus’ and ‘times’) versus the native environment of computers, base expansions. Borel’s question is essentially asking if there is a fundamental difference in ‘human’ mathematics and the mathematics of ‘computers.’
What are your favourite applications of your work?
I do mathematics for the sake of mathematics. I like solving problems and feel rewarded when I do. At times I have been influenced by Hardy. Like him, I subscribe to the idea that “Mathematics may, like poetry or music, ‘promote and sustain a lofty habit of mind’, and so increase the happiness of mathematicians and even of other people.” I cannot think of a better application.
What are some other areas of your field that are particularly interesting to you?
I’m interested in any good problem. But usually it has to have some arithmetic flavour. As such, I am really interested in problems in classical analysis. For example, a classical result from about one hundred years ago says that a power series with coefficients from a finite set that is bounded in a sector of the unit circle is a rational function. This means that an irrational power series with coefficients from a finite set has the unit circle as a natural boundary. Well, okay. But then this means something much stronger than transcendence. At least transcendence over meromorphic functions (this I showed with Yohei Tachiya), but what more can be said about such a function? Can it satisfy an algebraic differential equation? I would love to know more about this.
Why did you become a mathematician?
I actually first went to university as a chemistry major to eventually lead to medical school. I kept taking mathematics courses because they were an easy good mark for me. Along the way I realised I did not want to help people in the way a medical doctor could, and I realised that I was pretty good at mathematics. So I just kept doing it. Actually, until just a few years ago my parents were still asking me if I would give up this maths thing and go to medical school. I like to remind them that I am a doctor… just not a medical one!
Do you have any advice for future researchers?
“Wear Sunscreen. If I could offer you only one tip for the future, sunscreen would be it. The long term benefits of sunscreen have been proved by scientists, whereas the rest of my advice has no basis more reliable than my own meandering experience… I will dispense this advice now…” But really, all I have for advice are quotes from people I’ve heard over the years. Probably the best of these was from George Willis, my colleague at Newcastle, and even this was something that someone had told him: “Never give a problem too much respect.” It could be hard, but its a problem. Try. You could fail. I have on several occasions—many, but countably many so far. Also, get used to failure. It happens a lot more than success. Be it problems, job applications, grant applications, prizes, promotions, problems… you get the idea. But each success, no matter how small, is sweet. And worth it. As the saying goes, “Mathematicians… we’ve got problems.” And that’s part of the fun.
Deepening field knowledge and providing a networking platform, why are opportunities such as AMSI Summer School so valuable? What do you hope attendees take from your lectures?
Knowledge can be gotten anywhere, the thing that the AMSI Summer School adds is exposure to different people and other places. The first time I went somewhere new, for some reason I had this weird thought that whatever mathematics I knew would not be the way things were at that new place. I know this is stupid—really, it is in retrospect—but I had to go somewhere new to figure this out. Now I’ve been many places… and math is the only thing that seems to stay the same!
What do you consider your biggest achievement to date?
I’m a pure mathematician and for some reason I don’t quite understand, I get to solve problems and do mathematics everyday and somebody pays me for it. Seriously, I don’t really understand why, but they do. One of my PhD supervisors (Peter Borwein) used to tell me that, “An applied mathematician gets a job at a university and they complain and say that they could get a better paying job in industry and that the university should be happy they are here. A pure mathematician gets a job at a university and thinks, ‘You suckers! I’d do this anyway… and you’re paying me!’ “
Dr Coons is delivering the Analytic Number Theory topic at the 2019 Summer School.