# February 2020 Instructors of the Month, Terry Gannon and Vladimir Troitsky

Two professors with very different styles introduce two fields of mathematics to first-year students. Meet our February Instructors of the Month.

Andrew Lyle - 03 February 2020

Professors in the Department of Mathematical and Statistical Sciences, Terry Gannon and Vladimir Troitsky have very different instructional styles-each of which influence their courses in the analysis and algebraic branches of math courses. But in 2019, they tried something new: swapping their first-year honors courses, hoping to give students a more unified introduction to the world of mathematics and statistical sciences-and it was a great success. Hear from both of our February Instructors of the Month as Gannon and Troitsky share their thoughts on mathematics: what makes it like chasing squirrels, and why solving one problem often leads to discovering exciting new ones.

### What do you teach?

I try to teach people to think and play like a mathematician. Usually this is through a course in some area of algebra-at all levels, from first year to fourth year undergraduate courses.

Mathematics is an archetypal art: as old as thought, as human as song. Like any high art, it is the language spoken by nature.

There are lots of cool things. Nothing in art or science has as much science fiction in it as math, and there are lots of simple questions, which we still can't answer.

A number, n, is called perfect if the sum of its divisors is 2n, so for instance in the case of 6, 1+2+3+6=12 and 28, 1+2+4+7+14+28=56, 6 and 28 are perfect. Is there an odd perfect number? Are there are infinitely many even perfect numbers? No one knows.

"Mathematics is an archetypal art: as old as thought, as human as song. Like any high art, it is the language spoken by nature.." -Terry Gannon

Nevertheless, we are in the golden age of mathematics-old conjectures are being solved, and powerful new methods being developed now at a much faster pace than ever before. For example, in the last 25 years we've proved Fermat's Last Theorem and showed that you can't pack oranges denser than the way you'll find in any supermarket. Like all art, mathematics is all about cutting to the essence, stripping away the prejudices that cloud and distract. This century, mathematics will finally absorb the lessons of quantum field theory-which is the mathematically incoherent and experientially distant language of modern physics-and this will allow us to identify and strip another layer away from our built-in biases. I can't wait to see the new meanings of geometry, number, symmetry, and more that quantum field theory will lead us to.

But the coolest thing is the "doing" of math. Most of us might liken much of research to climbing a steep hill against a stiff breeze: every so often we stumble and roll to the bottom, but with persistence we eventually reach the summit and plant our flag amongst the others already there. And before our bruises fade and bones fully mend, we're off to the next hill.

But math research in its purest form is more like chasing squirrels, in my opinion. As soon as you spot one and leap towards it, it darts away, zigging and zagging, always just out of reach. If you're a little lucky, you might stick with it long enough to see it climb a tree. You'll never catch the damned squirrel, but chasing it will lead you to a tree. Mathematicians call these trees "theorems." The squirrels are those nagging little mysteries we write at the top of many sheets of scrap paper, puzzling thoughts that keep us awake at night. We never know where our question will take us, but if we stick with it, it'll lead us to a theorem, and if we're really lucky, maybe to a whole new structure.

That, I think, is what research in math ideally is like. Why can't we take the square-root of negative numbers? That of course leads to complex numbers. Why can't we divide by zero? That leads to projective geometry. In fact, mathematics is a prospector's dream: no matter where you squat and start to dig, if your eyes are open,your touch delicate, and your mind clear, you will find mathematical nuggets.

### What was your favourite learning experience as an undergrad? How do you incorporate that experience into teaching your students?

I loved it when the professor would digress-maybe talking about their own research or about something only vaguely related to what we were supposed to be discussing. It was exciting to learn that math was alive and creative and spills over boundaries. I love those moments when someone reveals to me a fresh perspective in something stiflingly familiar.

Each lecture I give, I want to show something groovy, something exciting. I want to blur boundaries, challenge biases. I want to show magic in places that look dry and dead. I want to get in the head of the person who discovered something and show that-on a good day, at least-we could have discovered it too. Certainly, my lectures are not always successful, and I fear that many students find my style frustratingly and confusingly nonlinear. But teaching is intimate, and an instructor has to be self-honest. I teach as I learned and as I continue to learn. That's why each student needs to sample a variety of instructors.

#### What was it that drew you to this field?

I find math truly compelling: I think-so I'm drawn to science. I feel-so I'm drawn to art.

### What do you feel is the most important piece of advice you give to your students?

Creativity and magic, like love, are things that happen when nothing else gets in the way. If you feel flat or bored, find out what is in the way and change it. If you can't see the stars very well, try turning off your flashlight.

### What is one thing that people would be surprised to know about you?

I'm a prairie boy. I grew up in small Alberta towns like Tofield and Viking. We complain a lot about education in this country, but my education was absolutely world-class-and that includes my undergraduate education, which was here at UAlberta.

### What do you teach?

I teach engineering calculus, honors calculus, honors linear algebra, and various graduate courses in functional analysis.

Calculus and linear algebra have applications in various areas of science, engineering, economics, and other disciplines. They form the foundation for many other areas of mathematics.

Honors courses introduce students to real research-style mathematics; they develop a deep understanding of the subjects. Students who want to really master calculus and linear algebra should take honors versions of these courses.

My own research is in functional analysis. I like this area of mathematics, and I encourage graduate students to try it.

In pure mathematics, one may spend months or even years working on a single problem. But when you solve it, you are the happiest person in the world. For five minutes. Then you realize that your answer leads to new problems-and this process never stops.

### What was your favourite learning experience as an undergrad? How do you incorporate that experience into teaching your students?

"One should really approach university with the mindset of getting an education, not only a degree. In the long run, marks do not matter; it's what you've learned that matters.." -Vladimir Troitsky

I had a high school math teacher who was an active research mathematician; he treated students with real respect, essentially as colleagues. That's also what drew me to this field-having outstanding teachers. I try to pass that on and do the same with my students.

### What do you feel is the most important piece of advice you give to your students?

Many students go to university to get a degree. They do not always understand the difference between "a degree" and "an education." One should really approach university with the mindset of getting an education, not only a degree. In the long run, marks do not matter; it's what you've learned that matters.

### What is one thing that people would be surprised to know about you?

I grew up on a Russian island in the Pacific ocean.