December 19, 2019

Mathematics whiz made significant advances in field of research

PhD graduate never thought he was skilled in mathematics

It seems difficult to believe, but Dr Lachlan MacDonald said he wasn’t great at mathematics in high school. He was, in his words, an “English and history person”.

Yet today (Thursday 19 December), Lachlan celebrated his graduation from the University of Wollongong (UOW) with a Doctor of Philosophy in pure mathematics.

Throughout his four-year degree, Lachlan has made significant inroads into a notoriously difficult field of research: foliated manifolds. 

A manifold is a nice smooth space, such as the surface of a sphere, and foliation involves breaking that manifold into layers, so they fit together in a neat, tangential way.

“I was studying the invariants of these things, called characteristic classes and what they tell you about the foliation you are looking at. The characteristic classes I was looking at were pioneered by two French mathematicians, Godbillon and Vey. Their theory tells you whether or not the layers of foliation are behaving chaotically,” Lachlan said.

“Imagine you are an ant, and you’re walking along these layers, walking along a path, looking at how the other layers behave as you walk along, how they twist and curve.”

While Lachlan’s research may be difficult for those who are not mathematically inclined to understand, his advancements in foliated manifolds have the potential to be used in artificial intelligence or high-speed computing in the future.

“At this stage, my PhD is purely mathematical, but we are now seeing parts of mathematics that were invented decades ago, which are extremely abstract and difficult to understand, find a use in data analytics,” Lachlan said. “We have such amazing computer power now, and mathematics is fundamental to that.”

Lachlan said he never considered that he was good at mathematics in high school. He had to work really hard at it, and felt much more drawn to English-based subjects.

He began his undergraduate degree at UOW in Medical Radiation Physics, and fell in love with calculus.

“I had to work really hard at mathematics at high school to get good results, which I think is the same for most people,” Lachlan said. “University mathematics is completely different to what mathematics is like in high school. I became really interested in calculus in my undergrad, and changed my degree to a double degree in physics and mathematics. My honours supervisor saw that I was interested in differential geometry, and suggested I start studying non-commutative geometry.

“It is a very powerful way of organising your thoughts about things,” Lachlan said about his path to mathematics. “It has a seductive quality that is so powerful because it could be used in all kinds of different problems. This is especially true now that we have such amazing computers.

“We are only beginning to understand the potential of pure mathematics. Much of it could still find an application in the future.”

Lachlan’s PhD co-supervisor Associate Professor Adam Rennie, alongside Dr Glen Wheeler, from the School of Mathematics and Applied Statistics, said Lachlan has made significant advancements in understanding the foliated manifolds.

Professor Rennie said very few mathematicians had made advancements in this field, and many of those who had were recipients of Fields Medals, which is mathematics’ highest honour.

Lachlan is humble about his achievements, however.

“I certainly wouldn’t put myself into that category of mathematicians. A lot of work has been done before me and I’ve been able to add to that in a fairly small way.”

Now based at Australian National University in Canberra, where he is undertaking a six-month postdoctoral position, Lachlan said he is thrilled to be graduating but it felt bittersweet to close the chapter on his time at UOW.

“I really loved every second of my PhD. It was three and a half years and it went incredibly quickly. I miss it already.”