Tyrone Ghaswala: The Journey of Math From Melbourne to Manitoba

By Jimmy Fryers

Tyrone Ghaswala is a PIMS sponsored postdoctoral fellow in the Department of Mathematics at the University of Manitoba. Originally from Perth in Western Australia, Ty completed his undergrad at the University of Melbourne before relocating to Canada to study at the University of Waterloo, where he completed his MMath and PhD. In 2018, he moved to Winnipeg to undertake his fellowship at the University of Manitoba.

Tyrone giving a talk at a University of Toronto TEDx event.

Throughout his time in academia, Ty has not only excelled as a mathematician but also as someone demonstrating the energy synonymous with the sun-drenched city of Perth. An accomplished squash player, guitarist, and regular volunteer, his CV demonstrates he likes to keep busy. He was named volunteer of the year while at the University of Waterloo, attended Math Circle sessions to teach math enrichment to high school students, and has won several teaching awards.

This is an extract of the full interview with Tyrone.

1. Why did you write your PhD on mapping class groups and covering spaces?

One of my supervisors gave me a problem very early on in my PhD in the area, as a starting point for something else in low-dimensional topology. However, I got sucked into the area, which I’m very grateful for. It’s a beautiful place to spend my time!

This image shows two sets of 3 curves on a surface with what Ty and colleagues called “mesh intersection.” What they prove is that if two sets of curves have mesh intersection, certain elements of the mapping class group defined by those sets of curves satisfy what is known as the braid relation: ABA = BAB.

An interesting part of this work is you can get a lot of intuition for what’s going to be true based on drawing pictures; if you draw your pictures clearly enough they constitute pretty convincing arguments by themselves. A lot of it can be visualized in three dimensions, which something the human brain is wired to do — this is somehow aesthetically pleasing to me.

2. Are there any real-world implications/uses for your research? If so, what are they?

None that I’m aware of. Just good old-fashioned advancement of human knowledge. The work I’m doing now is motivated by natural questions from work I’ve done before and. generally, trying to do mathematics that people care about.

In pure math especially, there are many more questions that anyone has time to answer and we have to decide what we’re going to spend our time working on and figuring things out that haven’t been figured out before.

The framework that we’re setting down, which has happened throughout the history of mathematics, is that pure mathematicians have done some mathematics and proved some theorems that they find interesting and seemed like very natural questions. These tools are sitting there waiting, and occasionally they’re picked up by people wanting to advance, say, certain technologies and realize that this piece of mathematics is useful.

A couple of years ago when the Nobel prize in chemistry was announced there was a big flurry of activity in the media around the application of topology in chemistry — it wasn’t discovered mathematically for the intention of helping out problems in chemistry. The more holes we have filled-in in an area the higher the chance that someone comes along in the future and needs exactly that.

3. If you were given an unlimited research budget, what would you like to work on and why?

If I had infinite time to do research and no pressure to publish, I would dive into 4-manifold topology. There are nice classifications of 2-manifolds and 3-manifolds, but a classification for 4-manifolds remains elusive. Recently there have been new insights into 4-manifolds in the form of trisections, which provide a link (in addition to those which already exist) between mapping class groups and 4-manifolds. It’s very intriguing to me!

4. You seem heavily involved in educational outreach, including participating in afterschool mathematics enrichment programs for high school students. What is your favourite aspect of teaching?

Definitely thinking hard about parts of mathematics that I take for granted. There is so much cool material in most undergraduate courses, especially in pure math. I always get a kick rediscovering a proof, or understanding something I really should have 10 years ago. More superficially, I really enjoy performing for an audience, so that aspect of teaching is also thoroughly enjoyable.

Ty Ghaswala, University of Waterloo. Presentation title: Pac-Man and Donuts

I’ve always been more comfortable on stage with an instrument in my hands growing up than I was on stage with just a microphone. I think getting a kick out of performing when I was growing up is what I like. That’s what I get out of it now.

Seeing students have an ah-ha moment and understand a problem is definitely one of the perks of teaching. There’re always parts of every course where this happens and it's very clear and very entertaining for them and, for me, it’s satisfying.

5. You’re an active speaker, giving talks numerous talks in various locations. What do you enjoy about giving these talks?

The most enjoyable part of the talks is usually what happens outside of the talk! There’s always a chance around the talk to chat with other mathematicians and exchange ideas. This is always an enjoyable experience, especially when you hear about an area you’ve never really thought about before. Occasionally someone gets interested in what I spoke about and we’ll have an enlightening conversation later on at dinner. This is one of the best things about doing pure math; it happens anywhere and anytime you let it!

6. How have you adjusted to the differences in life between Australia and Canada?

If you ignore the weather, the countries are very similar, so the main adjustment I needed to make was to buy a good solid jacket. That and finding ways to watch the cricket and Australian rules football are my main methods of adjusting. In fact, as we speak, I’ve got the cricket on in the background! Australia’s currently doing pretty well currently against Sri Lanka if anyone’s wondering.

7. It seems that music plays a large role in your life — even to the point of including fellow musicians in the Acknowledgements of your PhD. Why is music so important to you and how does it compliment your work as a mathematician?

Playing and performing music, especially with other people, is something I thoroughly enjoy, although it’s hard to describe why it’s so important to me. It complements my mathematics because it’s something I do other than mathematics! As our 80-year old clarinetist used to tell us over and over again, “you can’t be thinking about anything else when you’re playing music.”

I’m not deliberately saying to myself, “I’m stressed, I’ve just been thinking about math so I’m going to go and play some music.” It just happens because I play music so often. It’s a good way to be not consciously thinking about a problem.

Often in mathematics and music, you have to be stuck on a problem and have to be doing something else to figure it out. So this happens a lot: you’re stuck on a problem in the day, working on it, you have no idea what to do, making no progress on it, then later that day when you’re doing something else and are not consciously thinking about it, you figure it out. It happens over and over again.

Anything that takes my mind off where I was stuck is helpful, whether that’s having a nap, going for a bike ride or playing music.

8. What’s next for you?

I’m currently on the job market for another position, either a postdoc or a tenure track position. If I have my way I’ll continue in academia. Time will tell!

Contact Tyrone (as of February 2019)

Email: ty.ghaswala@gmail.com.
Office: 437 Machray Hall, University of Manitoba, Winnipeg, MB, R3T 2N2
Website: Visit the webpage

Publications (as of February 2019)

1. Promoting circle orders on a group to left orders, w Jason Bell and Adam Clay, in preparation.
2. Big Torelli groups: generation and commensurations, w Javier Aramayona, Autumn E. Kent, Alan McLeay, Jing Tao, and Rebecca R. Winarski, submitted, arXiv:1810.03453v2.
3. Free products of circularly ordered groups with amalgamated subgroup, w Adam Clay, submitted, arXiv:1807.08082v1.
4. Mapping class groups of covers with boundary and braid group embeddings, w Alan McLeay, submitted, arXiv:1804.10609v1.
5. Lifting homeomorphisms and cyclic branched covers of spheres, w Rebecca R. Winarski, Michigan Mathematical Journal, 66(4) (2017) 885–890.
6. The liftable mapping class group of balanced superelliptic covers, w Rebecca R. Winarski, New York Journal of Mathematics, 23 (2017) 133–164.
7. The liftable mapping class group, PhD thesis, Tyrone Ghaswala (2017). UWSpace. http://hdl.handle.net/10012/12075.

Awards and Honours (as of February 2019)

Teaching

• 2016 Outstanding Graduate Teaching Award, Pure Mathematics Department, University of Waterloo. For outstanding teaching by a Graduate Student as both a lecturer and a teaching assistant.
• 2014 Teaching Assistant Award, Pure Mathematics Department, University of Waterloo. For outstanding teaching by a Graduate Student in a teaching assistant role.

Academic:

• 2016 25th Anniversary FEZANA Endowment Scholarship for Excellence, Federation of Zoroastrian Associations of North America. For academic excellence.
• 2008–2010 6 time Wyvern Medalist, Queen’s College, University of Melbourne. For maintaining an average above 80% for the semester.
• 2009 Johnstone-Need Major Scholarship, Queen’s College, University of Melbourne. For outstanding academic achievement.
• 2008 Melbourne National Scholarship, University of Melbourne. An undergraduate entrance scholarship for academic achievement.
• 2008 A.L.Blakers Scholar, Australian National Mathematics Summer School. For the largest social and academic contribution to the National Mathematics Summer School by a student.

Extracurricular:

• 2012, 2014 Intramural Squash Champion, University of Waterloo. 2-time winner of the intramural squash competition.
• 2012 Volunteer of the Year, Graduate Student Association, University of Waterloo.
• 2009 Murray Sutherland award, Queen’s College, University of Melbourne. For outstanding contribution to the arts at Queen’s College.