New mathematics leads to innovative solutions

Laureate Professor Kate Smith-Miles is director of the ARC Industrial Transformation Training Centre in Optimisation Technologies, Integrated Methodologies and Applications (OPTIMA). She is also Professor of Applied Mathematics at the University of Melbourne and Associate Dean (Enterprise and Innovation) of the Faculty of Science.

One of the great myths about mathematics is that there is nothing new to invent. When I say I do research in mathematics, people say “don’t we already know everything about maths, isn’t it in all the dusty old books?” That’s because what people learn in school has been around since the 16th or 17th centuries, so they think “full stop! We’re done.” It’s not true – new mathematics is being invented all the time. Every time you are exposed to a problem that current mathematics can’t solve, you have to invent new maths.

I hadn’t planned to become a mathematician. I wanted to be a journalist. But in Year 12 at Hampton High School I had an inspiring maths teacher who gave me a glimpse, for the first time, of what mathematics is all about, and I started seeing neat stuff that I loved. She tried to talk me out of being a mathematician, saying “Where will it take you? You’ll just become a maths teacher like me.” I was horrified; I thought, “You’ve got the best job in the world, inspiring people.” I didn’t know where studying maths at uni would lead, but I thought they wouldn’t teach it if it didn’t lead somewhere – that’s how naïve I was! On my first day at the University of Melbourne it was clear to me I was going to be a university mathematics lecturer.

Kate Smith-Miles

I wasn’t exposed to research until I was awarded a summer vacation research scholarship after my third year. The project influenced my choice of an honours topic in fourth year, and then I did more summer work at the CSIRO because I was interested in real-world applications of research. I was already studying operations research, a field that emerged during World War II to help with critical decision-making, such as which troops should be sent to which battles, or how to decide the maintenance schedule for aircraft. They needed decisions that were provably optimal, not based on trial and error or best guess. Operations research is a field that applies to every area of life though, not just war-time decisions: what is the best sequencing for traffic lights, the optimal scheduling for aircraft at an airport, the best way to design a product?

With some of the questions we ask, there are too many choices to list them all and evaluate which is best. There are rules and goals, and we have to choose the best outcome from the many criteria we need to consider simultaneously. You can’t do it with a pencil and paper or a spreadsheet – you need powerful mathematical techniques to prove that’s the best you can do, you can stop searching now. These days I solve optimisation problems for Boeing in aircraft design, optimising windfarm operations for AGL – I have many industry partners and we’re always developing new techniques to handle their intricacies.

Practical problems have always appealed to me – that’s why I’m an applied rather than pure mathematician. I want to have an impact in the world, tackling problems that need innovation. Working with industry partners is a really good way to do that. Many mathematicians do research without knowing how it will be used, they are laying foundations. History is filled with examples of technology we have now because of some fundamental idea from 250 years ago. That poor mathematician way back then never lived to see the impact their work had. I don’t want that. I want to feel in my lifetime that my work has had some impact.

The best challenges are when an industry partner poses a question that we don’t immediately know how to solve, and the techniques we develop can be used far more widely. A good example is when Leica Biosystems wanted to improve their pathology testing machines. If, for example, they are testing for melanoma, the tissue sample is sliced thinly to go on glass slides. Each of those slides goes into the machine, which reads the barcode and knows the recipe it has to follow. This involves a complicated sequence of actions carried out by robots.

Our job was to sequence the actions to maximise throughput while minimising energy consumption and avoiding wasting expensive chemicals. We found that the mathematical scheduling problem had some symmetries that could be identified and eliminated to make the calculations much faster. That technique can be applied to speed up scheduling algorithms in other areas. I loved that project. It was fantastic to feel our mathematics was helping improve pathology testing, there were societal benefits, there were lots of mathematical challenges, and it led to a contribution to knowledge more generally.

The issue of algorithmic trust is something that has occupied my research agenda for much of the last 15 years. Society has a big challenge ahead as we become more dependent on algorithms making decisions by themselves, whether machine-learning or self-driving cars. As humans, we have to be able to trust them, and one of the reasons we can’t at the moment is the way they are tested with example problems or scenarios. How many scenarios are enough – 20, 50, 100? My research focuses on developing a methodology by which you can know you have tested an algorithm comprehensively and rigorously with a sufficient number and type of scenarios. I have a website – – where people can develop visualisations to see which examples they should be testing their algorithm on to understand strengths and weaknesses.

My high school Year Book when I was 17 quoted the ambitions of the graduating class. Many classmates said their ambition was to be a millionaire or own a yacht. My ambition was to develop Kate’s Theorem and have someone else in the world use it. I was inspired by Pythagoras’ Theorem, and its mathematical proof that shows it is always true. I liked the idea of my own theorem but just as important was the idea that someone would use it. I still think that was a pretty good goal, and it’s still my driving ambition. There is no Kate’s Theorem of course – it was naïve at 17 to think that anyone other than greats like Pythagoras could have that honour – but I do have a whole heap of mathematical work that other people use, and that’s plenty satisfying.

Among many awards, Kate has won the 2019 Renn Potts Medal for outstanding contributions to operations research, the 2017 EO Tuck Medal for outstanding research and distinguished service in applied mathematics, and the 2010 Australian Mathematical Society Medal.

As told to Barney Zwartz. Photograph of Kate Smith-Miles by Michael Hood.

The University of Melbourne has announced the establishment of two new major investment funds dedicated to supporting Melbourne’s world-leading researchers to turn their extraordinary discoveries and innovation into commercial reality. Find out more about the University of Melbourne Genesis Pre-Seed Fund and Tin Alley Ventures.

First published on 12 January 2022.

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