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23-year-old Advanced Mathematics student Peter Ayre opens up about the highlights and challenges of being an Honours student, ponders on what he'd be doing if he hadn't pursued maths, reveals his plans upon completion of his degree, and tells us what initially drew him towards maths (and what continues to keep him engaged).

Interview conducted by School Communications Officer, Susannah Waters

Why mathematics?

The reason that I initially went into mathematics was because when I went through high school, it was always something I could just sit down and do. Writing essays seemed laborious, and it was actually painful to get me to go through that kind of thing. But when you do maths, you kind of just sit down and do problems, and you get little wins when you work things out. It was just easy to study.

Part of the reason I stuck with maths is because as you go through, it changes a lot. As in, you do maths in high school and you’re solving equations and things like that, and then you go to early university and things start becoming abstracted and you’re living in this new world. And then you get to later undergraduate maths courses, and it’s completely different to anything you’ve ever seen; it just constantly keeps changing. It’s just continuously engaging.

Whereas I think if I studied something else… you kind of get introduced to a field and you’re getting more knowledge about the field, but I’m not sure that it changes on that kind of level. I think that this changing is what has kept me engaged throughout the years. Mathematics also has this nice little community that you buy into when you start as an undergraduate, and it’s definitely something that keeps you coming back.

How old were you when you realised you were interested in maths?

Probably my earliest memory of my interest in mathematics was on some ridiculously long family road trip. I was asking my mum for numbers to multiply together… in the car – just for fun.

So, evidently, I was kind of an odd child! But it’s something I always found easy to do, and I don’t think it would have been like I was X years old when I became interested in maths. I think that as I’ve gone on, I’ve become more interested. With regards to what I was saying before about mathematics changing a lot, I think that’s a big part of why my interest has maintained. Whatever stage you’re at… two years down the track you’re probably going to be doing something that is very, very different. And it’s whether after that change, it is still interesting for you.

If you could wake up tomorrow and be any mathematician (currently alive or from any time in history), who would you be?

People will tease me about this, but I’ll say [Paul] ErdŐs. Not because I’m particularly envious of the fact that he was a drug addict, but more the fact that he was very big on the idea that maths is something that isn’t you sitting in a room, it’s something that’s done collaboratively where you work with your peers - and that’s how it’s supposed to be done. He was very big on pushing that agenda for maths, and I think that was really important and is one of the best parts of maths. So in that sense I’d probably pick him.

What is your ideal job?

I’m a bit strange in that I study maths and I quite enjoy it, but I hate working with numbers and I don’t particularly like statistics. I’m very much on the theoretical maths side and I enjoy it because the problems that you get and are faced with are really challenging and interesting to work with. Pure mathematics is quite different to working with data and doing analysis with statistics. And I think statistics is more for people enjoying finding an answer, or revealing some fact. For me, it’s more about the problem itself and solving the problem. So I think academia is obviously one thing that’s a possibility. I’m already part of the way there.

But other than that, I’m taking a job next year in management consulting. Which I think could be an interesting place to work – you get a lot of interesting problems and I guess that’s what I was trying to allude to before: it’s not specifically this abstracted mathematical world that I’m interested in, it’s challenging problems and ideas more generally. I think - hope - management consulting is a job where I would get those kind of problems – albeit with maybe different qualities than academia.

So you’re not moving straight on to a PhD?

Not straight away, I’m doing Honours at the moment and it’s a lot of work so a break from study will be nice - and also a chance to do some travelling!

Maybe in the future?

I’ve just had this conversation with my supervisor! I have this unnerving feeling that a few years down the track, I will succumb to my romanticised view of mathematics and enrol myself in a PhD. We will see!

What's it like being supervised by ���ڳ��ܰ�dz���𱹱�?

He’s very nice. He’s hands off in the sense that you get to do a lot of your own work, so the work that you do is in some meaningful sense your own, which I’m not sure you can get all the time – if someone gives you too much help then solving the problem is stolen from you! Nonetheless, Fedor always lends a guiding hand and prods me in the right direction, letting me actually get there by myself.

Let’s imagine for a second that you hadn’t pursued studies in mathematics. What would you be doing instead? Or is this a prospect too painful to contemplate?

I kind of came into university with the idea that I would do maths and physics, so I think naturally it would be physics. I don’t know if that hugely answers the question, because physics is to some extent the same thing as maths…

So let’s pretend maths and physics didn’t exist!

Yeah, ok – I think if I hadn’t pursued maths I would probably have done commerce. I think that doing a degree which I actually enjoy has been a vastly different experience - to the extent that I even identify as an individual with being a mathematician. However, I think if I did for instance, commerce, I wouldn’t have been as engaged with my studies – I would have missed this whole experience and would be a different person for it. When I went to do mathematics at university, I went to do research-style of work, and inadvertently grew up, academically, in that kind of environment. If I hadn’t done maths, I actually have no idea where I would be at this stage.

So I guess you’re happy that maths exists?

Very, very happy.

What things do you enjoy doing that help you switch off from uni work?

I like getting outdoors, rock climbing, I go to the gym, I have a motorbike, I play tennis – just the regular kind of things, nothing particularly fancy. When you’ve been sitting inside all day, you need to get out and just move around and do something. You go crazy if you don’t get out and move around a bit.

You’re currently doing Honours. Tell us a bit about your Honours thesis – what are you focusing on?

The area that I’m doing my thesis in is Functional Analysis – and more specifically than that, Operator Theory. If I tried to explain anything more specific than that, it probably wouldn’t be comprehensible to a lot of people!

I’ll attempt a vague description. In high school and in early university you spend a lot of time studying functions either on the real line or on the complex plane. There is an area of mathematics where these functions can be applied to extremely abstract objects in a sensible way; my thesis is looking at how certain properties of the original functions - continuity, differentiability, etc. - translate into this more abstracted setting. Taking a moment to think about it, the properties that I mentioned before are defined in terms of the underlying space, and so many of the results we seek aren’t so apparent.

Regarding your experience as an Honours student:

- What have you found wonderful?

I think the nicest thing about doing Honours in maths is that it’s the first time that you properly really get involved with the School of Mathematics and Statistics on a community level. As there are only about 5-10 people in your class, all your lecturers will know you by name and are all quite familiar with you, to the extent that they know how you think and how you solve problems. They are actually meaningfully concerned with your work – they’ll go out of their way to attend your seminars, and talk to you about their ideas and how you could change things in your thesis and things like that. When you become an Honours student, the involvement with the school on a communal level is really nice and it’s probably one of the best parts of being an Honours student.

- What have you found challenging?

The most challenging part of Honours is that you do coursework throughout the entire year, and with your coursework you have really immediate obligations – like an assignment due in two weeks, lessons that need to be revised; but you’ve also got a year-long thesis that’s looming in the background.

You really need to manage your time between the two, and it’s quite difficult in knowing what pace to go at with your thesis so that you put enough focus on your coursework and vice versa. It’s the first time you really have to properly manage these kinds of obligations. The thesis is like this huge product that comes out at the end of your Honours, and it’s not the easiest thing to balance. For instance, when you’ve got a lot of assignments, you often need to actually force yourself to stop working on them because other things need to be done. You can’t let it consume you.

- What have you found surprising?

I’ve got a surprising instance rather, which kind of has bearing on the whole Honours process. I did my seminar, which is the practice that we do for our big talk we do at the end of the year, and one of my friends’ supervisors came to see her speak. He ended up watching mine as well because we were speaking straight after one another, and he actually approached me and said he had some ideas on how I could alter my result. We then walked back into his office and talked about it for at least an hour or two. My supervisor is obviously very involved with what I do, but for someone else who has no obligation to be involved in my work, to pull me aside and spend their time helping me, it was really nice.

Actually the other surprising thing – this one’s probably more surprising, maybe the other one goes in the wonderful category – is that you spend all this time coming through undergraduate maths and you’re like this stupid little child compared to your all-knowing lecturers. They know every result off the top of their head, and by comparison you’re essentially just stumbling around in the darkness. But then you get to Honours and at least part of the way through the year you’re kind of getting to the point where you’re at research-level mathematics, the cutting-edge of things, and you can have an intelligent discussion with your supervisor about research-level mathematics. I found it really surprising; I thought there’d still be this massive gap where I’ll go through a PhD and halfway through maybe – maybe – I’ll see the light at the end of the tunnel. But you get there surprisingly quick.

I think it’s because when you do Honours you pick a really specific area, and you immerse yourself in that specific area, but it allows you to get quite far - which is something that I never could have appreciated before I did it.

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August 2013