# Exploring the marvels of mathematics

## Professor David McKinnon shares his perspective on the beauty, purpose and fun of pure mathematics

David McKinnon, chair of the Department of Pure Mathematics, is known throughout the Faculty of Mathematics for his sense of humour and enthusiasm for teaching.

McKinnon, who has been at the University of Waterloo since 2001, has received numerous recognitions for his commitment to education: a Faculty of Mathematics Award for Distinction in Teaching (2008), a Distinguished Teacher award from the Centre for Teaching Excellence (2017) and an Award of Excellence in Graduate Supervision from Graduate Studies and Postdoctoral Affairs (2023).

He is passionate about encouraging innovative math teaching in the Faculty of Math as a whole, as exemplified by the Math Teach-Off he organized this March along with Statistics and Actuarial Science lecturer Diana Skrzydlo.

McKinnon is also excited about sharing his love for the beauty and fun of math with anyone who wants to learn more. The following interview has been edited for clarity and length.

What exactly is “Pure Math”?

The phrase “Pure Math” feels a little bit condescending to the rest of the maths — like, we’re pure, the rest of you are impure — but what it means is that it’s just math for the sake of math.

If you’re doing mathematical physics, then you’re doing physics with a lot of math in it. If you’re doing applied math, then you’re doing the math but the questions that you’re trying to answer come from somewhere else — physics, medicine, climate science and so on. But in pure math, the questions you’re trying to answer are from some other part of math.

I love math because math is something that you can be 100 per cent sure of.

What do you specialize in?

Arithmetic geometry. Basically, taking questions about numbers and number theory and using geometric techniques to answer them.

For example, in arithmetic there’s something called a Pythagorean triple: three positive integers that could be the sides of a right triangle, as seen in the Pythagorean theorem. For example, 3, 4, and 5, because 32+42=52. So, we can ask arithmetically which other trios of numbers are Pythagorean triples.

Now, it doesn’t seem obvious at first why the two are related, but there are connections between whole numbers and circles. So, the geometry of circles is connected to the arithmetic of Pythagorean triples.

Right now, I’m looking for a similar kind of connection between points with rational number coordinates and certain kinds of curves. I have a guess — a conjecture — but I’m trying to prove a pattern.

There is beauty in the structure – in suddenly understanding those connections. It’s like standing in a forest, and thinking you’re just looking at trees, and then realizing it’s an orchard! Suddenly, you see the pattern.

It’s the same thing with my research. There’s structure there. It’s not trees, but ideas, and it’s not an orchard, but a theory, yet the way that things are related to each other is analogous. When you see the orchard, you can appreciate the patterns in it.

Why did you choose this career?

This always comes up at parties. First people ask, “what do you do?” and then their follow up question is usually either, “Oh, I wasn’t very good at math in high school,” or else maybe, “What is that good for?”

And there are answers to that question — there are reasons why I get paid to do this. It is useful, and historically has been useful for humanity. And we don’t always know, in research, what will eventually be useful when we do it. A certain portion of pure mathematics will be useful someday — but if you don’t spend money to do it, it won’t get done, and then you will never know what you missed.

But that’s not why I do it.

People don’t ask NHL players why they chose their career, but my answer is kind of the same: I do it because it’s fun! Now, they do pay the NHL players way more! But I think our motivation might be the same. Someday, my research might be useful, and along the way I can teach a lot of kids how to do math — but I do this because it is beautiful, and it is fun.