Welcome back to part 2 of our interview with Kieran Cooney! We discuss participating in Mathematical Olympiads and self directed learning. If you missed part one be sure to check it out here .
Did participating in Mathematical Olympiads help you in college and with your career?
I mentioned previously that it made me aware that there was something out there that I loved. One thing that it gave me was educational independence.
I was in these enrichment classes for three years. The first year I didn’t do well in the Irish Maths Olympiad, in the selection test. In the second year I worked quite hard and I made it onto the team by the skin of my teeth. But leading up to that I was studying quite a lot and to be quite honest about it I was sick of the Leaving Cert. I don’t mean to be demoralising about it in any way but it was just my own opinion, I hated that thought every morning of going into school and I missed asking myself ‘what am I going to learn today?’ because the Leaving Cert is more about getting your points.
So I had discovered something, I discovered maths, I discovered competition maths, and I loved it and I loved learning it and it was fantastic. It was at my own pace, I’d be at breakfast and I’d have this book on discrete mathematics while I’d be eating. I’d be reading through it and solving problems, and that’s something I still do. Can you imagine then, you have this person who is reading maths books and is devouring them, who can’t get enough of them, and then you send them to college where they have a library? You can go in and you can read as many of these books as you want. I was over the moon. My first week there everyone else was going to the pubs and I was in the library. There were just so many books, it was amazing.
That’s something that’s carried through to today. I love just picking up a book and going through it. I have a systematic way of going through things and taking notes and really developing an understanding of it, because particularly in the Olympiad it was always encouraged not to just develop a surface level understanding of something. Go through the problems, understand the problems, make sure that you can replicate the arguments in this book yourself. I’m able to independently think for myself, is really what I’m trying to say in summary.
Also, of course, general problem solving techniques too. There’s nothing quite like being given three maths problems and being told ‘well you’re not leaving here until you solved them or until after 4.5 hours, so take your pick’. That will really sharpen your knife when it comes to thinking outside the box and problem solving. And that’s really stood with me, definitely. When I was doing my masters, that was a research project, and it was basically just solving a problem for a year by myself and I don’t think I would have had the techniques necessary to do that without my Olympiad experience.
Could you share some of your techniques for reading maths books?
It’s kind of difficult for me to say. I think the key is to enjoy it, that’s not something you can learn. I think you can learn what you enjoy and then pursue that. As for me, I’m kind of cheating because I like it. I think the most important thing to figure out about self-driven learning is the balance.
There’s this really famous geometry book in contest maths, Geometry Revisited by Coxeter. Check it out. If you like geometry you need to read this book. I was told ‘make sure you solve all of those problems and don’t move on’, and that’s what I did. I was doing this during college actually, so I was punishing myself more than preparing for any contest, and I was stuck on a problem in it for months. And eventually I solved it and I was very proud of that balance between figuring out myself. But what if I was stuck on that for years? I would have not been productive anymore.
There’s that balance there between figuring out when is the right amount of time to spend understanding something, when do you move in, when do you decide to go off on a tangent and research something else, when do you realise that actually you’re bored of this and you’re not quite motivated. Maybe you’re pushing yourself on to do it even though you don’t want to, but it might make more sense to move on and learn something else.
Particularly if you’re on your own doing something, and you’re not doing a course online or you’re not doing a class, making sure that you stay motivated and productive is crucial, and to make sure that you’re still enjoying it. Because at the end of the day you’re doing is for you, no one else.
How long do you normally spend reading a maths book?
It depends on the book, it varies. Maths books are beefy. If you’re reading fiction, or a novel, or prose, or whatever, if you’re reading a paragraph you’re reading a paragraph, but in a maths book, reading a paragraph, you might have to read the same thing a dozen times before it sinks in, and then put the book away, and come back to it a week later, and read the paragraph again. It’s just it’s much more dense, and that’s the main struggle.
But what I will say is as you do more of that over time it gets easier. Number one because you’re getting used to reading technical books and number two because like a lot of maths is centred around these core concepts, functional analysis, algebra and combinatorics. As you start to learn one you see ‘oh well I’ve seen this before, it’s kind of a similar idea’.
And so to give you an example, this is one book I’ve just finished, Pattern Recognition in Machine Learning and this thing is dense. I’ve been keeping this by my side while I’ve been working, so if I’m training a model I’ll read this. My approach to this has been kind of interesting because there’s been some chapters of this that I’ve been very in depth about and I recognise ‘oh this is interesting, I can get something from this’ and there’s been other chapters that have been very opaque, and it wasn’t clicking with me. Rather than fighting it I just skimmed through the main points and I said okay. And so overall that took me a couple of weeks.
I was reading another book before Christmas, Analytic Combinatorics by Flajolet. It’s an interesting book on applying calculus to solve combinatorics problems and that took me maybe three months to get through two chapters out of ten. I gave up on that one but I still got a lot out of it.
What maths books would you recommend?
I have one that I have been recommending to everyone for years. Not only is it one of my favourite maths books, it’s one of my favourite books ever. I think it’s terrific. It’s called Visual Complex Analysis by Tristan Needham.
He is a lecturer in San Francisco, and I first heard about the book when I was in the Olympiad actually. We had a guest lecturer who was teaching us about an inversion in a circle, and applying it to geometry. He said this book is really interesting. And so that first week that I was in college, I went straight to find this book in the library. And I read it. It’s just it’s really, really well written. You wouldn’t need a degree in math to understand it, but you would want to be maybe halfway through.
I would recommend giving it a go. Chapter one is really, really approachable. And the problems are really nice. The writing style is like nothing you see in any other maths books. It’s so conversational and it explains things so well. It really broadened my understanding of mathematics in general. There’s some physics in there, too.
Good news, he’s been promising us his next book, Visual Differential Geometry, I believe it’s called. He’s been promising us that for years and he never got around to it. But he just announced lately that that will be coming out later this year. Check it out.
Do you have any advice for young people interested in STEM subjects?
I think it’s really important to do what you like. You see so many people these days who don’t really settle into a career for many, many years because they decide actually, no, I tried this for a couple of years and I don’t particularly like it, so I’m going to go back and do this. Do what you like.
Staying motivated is a really important part of work and college. If you’re not motivated to do something, you’re not going to have a good time. So, try and find that thing and stick to it. In Ireland, we have a tendency to specialise quite early and something I saw in America is that they studied quite a lot of things in college, like you could be studying physics but you’d have to take a mandatory language class. I think that’s a very good idea.I would recommend people to, early on, while you can, try and study a wide range of things if that interests you. It’s really important.
That expression comes to mind to try and know a little about a lot and to know a lot about a little. Having that breadth of knowledge is very important. Something in college that I didn’t do, which did cause me some problems, is that I didn’t learn a programming language. I only learned it later and it was a struggle for me. Whereas now that I’m a data scientist, it comes no problem to me at all, this thing that I was quite frankly afraid of. The only time that I learned it was actually when I got a job doing it.
My advice there would be to try and make it a priority to learn some kind of programming and don’t let it become that thing that you’re not going to do. Don’t define yourself by that, because it’s so important both in terms of industry and, I think, in academia. I think it’s something that’s kind of underused in a lot of different parts of academia, even in pure mathematics.
Using programming, you can automate things, you can run calculations, you can make nice pictures. So many different things, it really is the language of the modern world now. It’s a shame not to be able to make the most of this wondrous toolbox you’ve been given. When you put the time in, it’s not that hard, especially compared to contest mathematics. It’s not as hard as doing an IMO. I would put some effort into it.
Thank you very much to Kieran for sharing his thoughts with us. You can check out Kieran’s Linkedin .
This article is transcribed from a spoken interview.