“Simple, Yet Powerful”: Mathematician Jane Breen Wins Distinguished Dissertation Award for Work with Markov Chains

The relationship between a graduate student and a supervisor is a tricky thing. While it’s crucial for the student to have sufficient space and time to investigate their own interests, it’s equally important for the supervisor to ensure their protégé isn’t wasting precious time and resources chasing dead ends.

Jane Breen (Ph.D., Math ’18) considers herself particularly fortunate when it comes to her former supervisor, Department of Mathematics Head, Dr. Steve Kirkland. Not only was their working relationship an unqualified success, Breen’s doctoral thesis (entitled “Markov Chains under Combinatorial Constraints: Analysis and Synthesis”) was awarded the UofM’s Distinguished Dissertation award.

Offered by the Faculty of Graduate Studies to a graduate student in each of six areas (applied sciences, health sciences, humanities, interdisciplinary sciences, natural sciences and social sciences), the D.D. award recognizes a groundbreaking piece of original work. Unsurprisingly, Kirkland has all good things to say about Breen’s thesis:

“One of the striking features about the research in Jane’s thesis is the diverse range of mathematical techniques she brings together to establish her results. Applied probability, generalised inverses, graph theory, nonnegative matrix theory and numerical analysis are all part of her tool kit, and she uses each to good effect in the thesis.”

“A good deal of mathematics is centred on problems and solutions, but mathematics is also a language, and Jane has that aspect in hand as well. Her thesis is remarkably well-written – she has an engaging, lucid style that is a pleasure to read.”

Breen’s opinion of her former supervisor is no less glowing:

“Oh, he’s a really excellent supervisor. I have nothing but the highest praise for him. I kind of reflected on this just towards the end of my studies, because I nominated him for an award because – you know, you get to the end and you don’t know how to show your gratitude to somebody, and you think everybody needs to know what a great supervisor he was. I wasn’t very experienced. So, he kind of gave me a lot of assistance in the beginning. As the years went on, he made sure that I was given space to come up with ideas by myself, and given space to work on things, and figure things out, and choose my own direction as well.

“That’s something that’s hard to strike a balance with, you know, when you’re supervising a graduate student, I think. He’s also so generous with his time. He was the head of the department when I came over to the University of Manitoba. I’d meet him once a week, sometimes twice a week, and he’d always have time for me to talk about what was going on, what we were working on.”

What they were working on was Breen’s favourite mathematical model:  Markov chains.

“What I like about it is that it’s a really simple mathematical model, in that it’s quite basic and it assumes very little about what you’re trying to model. It’s just some basic kind of things where you assume that you’ve got a system that can be in any number of states and at any given time it can transition to be other states. It kind of moves around. So, it’s quite simple, but it’s also quite powerful, and you can take things that are very, very complicated and then model them with this kind of nice, simple system and really say some big things about what’s going on.”

Mathematics was just one of the subjects that Breen enjoyed studying in school. She admits that she never thought there was anything special about liking math; that it wasn’t until high school that she understood her aptitude for the subject was out of the ordinary. It was at that point that one of her teachers told her something that changed her outlook.

“My math teacher was a great teacher. She said something to me like ‘you know, some people work really hard at this, but you just have it. You have that something special’, or something like that. Actually, that’s an attitude that I try not to encourage in my students. Because students come into university classes and they think ‘math is just something that I don’t get, so I’m never going to be any good at it’. So, it’s not an attitude that I actually agree with. But that was the first time I thought that ‘Oh, okay. Maybe this is something special and I should pursue it’. I always loved it, but I also loved English and art and biology and everything. I’ve always liked everything that I’ve tried. I realized then that being interested in math was something special and I should pursue it a little bit further.”

Despite the fact that no one in her family was particularly mathematically inclined, Breen’s success in the subject was undeniable.  After completing her B.Sc. in her home country of Ireland at the National University of Ireland, Maynooth (now Maynooth University) in 2012, she met with Kirkland to talk about the possibility of doing graduate work under his supervision. Kirkland recalls the meeting clearly:

“I told her a little about the mathematics that underpins an algorithm for ranking web pages, as an example of the kind of topic she might work with in her grad program. At the time, I was struck by how quickly she picked up on the main ideas – it was clear that she had a natural affinity for the subject.

“When I decided to move to the U of M in the fall of 2013, I really encouraged Jane to come to the UofM so that she could continue her studies with me. She showed terrific potential, and I definitely wanted to keep working with her!”

Breen was faced with a potentially life-changing decision:  switch to a different supervisor or move to a new country to continue working with Kirkland. Ultimately, she decided to take the plunge, moving to Winnipeg in the depths of a particularly frigid Canadian winter.

“I arrived in January 2014, in the middle of the worst winter that they’d had in 150 years, I was told. (Laughs) It was fun. It was kind of an adventure, you know. We didn’t realize at the time that the weather was unusual for Winnipeg, so we would go out in it all the time, when all the people from Winnipeg were staying in. It was minus 30 most days, and we were just going out and exploring.”

Breen settled down to her study of Markov chains once more. For the uninitiated, a Markov chain is a mathematical model which describes a sequence of possible events in which the probability of each event depends only on the state attained in the previous event.

“One of the reasons I particularly like Markov chains is that you can model a really wide range of things with [them]. Two of the things that I’ve thought about during my research – one of them has been a road traffic network model, where you’re modelling how the traffic moves throughout a network, based on turning probabilities and things like this. Another one that I’ve thought about has been molecular conformation dynamics, which I know very little about. It’s something to do with how molecules can kind of change their form. They can still be the same molecule, but they’ll change from one folding of it to another over time, and this kind of thing. It’s something that’s really important in drug design, when you take a drug, it can change its’ form once it’s inside your body, something like that.”

“What I like about Markov chains – the work I was doing you know, it was taking a model that could describe these two completely different things, but then I’m just working on the mathematics of it, and then anything that I do in the abstract setting could have implications in these really different settings. Part of it is as well that back when I was in school, and I enjoyed every subject that I worked on – this is kind of a way for me to still be involved in lots of different things. My device is kind of fundamental to a lot of stuff, so this is kind of a way that I can work in a lot of different areas while still sticking with one thing.”

Currently a post-doctoral associate at Iowa State University working under the supervision of Dr. Steve Butler, her focus has shifted slightly to spectral graph theory, an offshoot of her Ph.D. research. She is working with a group of undergraduate students, which she says has given her some insight into how things might have been for her own Ph.D. supervisor.

“I’m also learning a little of bit of what it must have been like for Steve to supervise me right at the beginning, too. You know, when somebody comes along and they don’t really know how research works, and you want to guide them and get them inspired and get them going by themselves.”

Asked where she sees herself in the future, Breen says she hopes to continue branching out on the research she did while at the UofM.

“I never initially planned to move to Canada, but now it’s become kind of a second home to me and I don’t see myself leaving any time soon. I’d love to be in a position where I can mentor undergrads and graduate students. Be involved in more research that’s kind of – this interdisciplinary stuff where you can work in mathematics but have it apply to a bunch of different things at once. Something that I’m really passionate about. Keep teaching, keep researching and keep being involved with other people.”

Author’s Note:  At press time, Breen advises that she has secured a position at Ontario Tech University as an Assistant Professor. She is looking forward to starting there later this summer.