*Benoit Charbonneau is an associate professor of Pure Mathematics who has been at Waterloo since 2010. We sat down with him to talk about his research and administrative work, his love of geometry, and why he does what he does. The following interview has been edited for length and clarity. *

## How did you decide to pursue a career in mathematics?

I grew up in Québec, east of Montréal, and in my fifth year of *école secondaire *– what we would call Grade 11 in Ontario – I had a fantastic chemistry teacher. I thought science was wonderful, so I thought that I would also like to be a chemistry teacher.

I was good at math too, though, and had been competing in math contests since I was in elementary school. I did well – well enough that I got to represent the school at later competitions. When I was in CEGEP [which Québec students attend after secondary school either for technical training or before university], a professor, Fernand Beaudet asked some of us if we wanted to learn some university math over the summer. So that summer, that’s what I did: I worked at a swimming pool store, and I learned mathematical analysis. That summer was when I realized math could be fun – even more fun than chemistry! I remember the exact day I decided I wanted to study mathematics: October 3, 1994. I ultimately attended the Université du Québec à Montréal for my bachelor’s, and I did my master’s there as well. I then did my PhD at MIT, then a post-doc at McGill, then after three years working at Duke I ended up here.

Essentially, I decided that I was going to keep doing math until no one wanted to pay me – and they still pay me to this day.

## What is your research about?

Pretty much everywhere, there’s this sort of feeling that applied math is separate from pure math. That’s not the view I have – probably shaped by my time at Duke, where there were a lot of people who were doing both. You’ll see that in my CV as well – I’ve done both, and collaborated with many people who were not in my field, particularly soft-matter scientists. We were able to put our two specializations together to advance something.

My thesis, however, and much of my research is on “gauge theory,” which I define as the general study of structures on vector bundles or principal bundles. In the field of physics, researchers work with gauge *theories*. We are both working on similar problems, but sometimes using different approaches and tools. Mathematically, I’d say gauge theory is at the intersection of differential geometry, algebraic geometry, and mathematical physics.

Classical geometry isn’t my main field, but I love playing with it. In 2019, I organized the Great Polytope Barn Raising Project to build a physical model of an omnitruncated dodecaplex, a four-dimensional object. On January 25, 2019, volunteers spent fourteen hours working to assemble the model in our M3 building. Unfortunately, we ultimately had to give up on our plan to suspend it from the ceiling because it could not support its own weight! The math worked; the physics just didn’t! Considering this a prototype, we are hoping to rebuild at some point in the future.

## What do you do here at Waterloo?

Right now I am on sabbatical, but usually I wear two different hats. I am an associate professor of Pure Mathematics, and other than this year, I have been the associate dean for undergraduate studies since 2019. Obviously, those were some tough years to fill that role, with the pandemic, and things have changed. We are more aware than we used to be of things like mental health, accessibility, and other equity issues.

I brought to the role a certain amount of unusual experience that helps me to tackles these problems: I was chief negotiator for FAUW (Faculty Association of the University of Waterloo), and worked on salary anomalies and salary structure, and also coordinated the adoption of Crowdmark, the system used on campus for distributing grades.

Another thing I’m proud of is the Bridge Lecture Series, which unfortunately was put on pause by the pandemic. These were public lectures connecting arts and humanities to science and mathematics. They would be led by two experts from different fields, who had to deliver one unified lecture together about a topic.

The idea came to me back when I was a math professor at St. Jerome’s [from 2010 to 2014], and I was regularly having lunch with colleagues from multiple disciplines (history, English, religious studies, etc). I thought, “let’s make something really interdisciplinary.” I know that math is scary for many people, and they will not participate in math-centered things. But I heard from a lot of people after who said they would not have gone to a math lecture, but because there was only a little math, they went.

Once we had a Tiffany glass artist, and a geometer talking about Platonic solids. Probably the most popular was with a historian and a mathematician: the historian talked about D-Day, and then we had a real Enigma machine and explained how it worked. You’d have people come from all over the place, both to give the lectures and to listen. I always felt like I learned something too. And it felt like outreach.

## What do you wish people understood about mathematics?

When people ask me what I do, at parties or what have you, I compare pure mathematicians to cartographers. We’re pushing back the frontiers of ignorance. Say you get to a new island, and you have no idea what’s out there, so you kind of explore. You create a map of what you find, even though you don’t know what is useful. You might find sand, or something that stinks and you don’t like it – and then years later, people realize what you found is an oil sand, and it becomes useful. We don’t know when we discover things whether they’ll be useful.

In pure mathematics, I get to play with things that people have developed over millennia, simply because they were curious. Sometimes we find usefulness, and sometimes we don’t. It’s sort of the wrong question to look for useful things. Before knowing what’s useful, you need to know what’s out there.