Mathematical Discoveries and Connection: Introducing Professor Catherine Greenhill (’87)

Catherine is a Professor in the School of Mathematics and Statistics at the University of New South Wales, having published more than 50 research articles answering questions at the interface of mathematics and computer science.

For those who could not join us, below is a copy of Catherine’s speech.

“Thank you very much, Ms Curtis, for the introduction. I really am very honoured to receive this award.

As you have heard, I am a mathematician and a Professor at the University of New South Wales (UNSW), Sydney. I’d like to tell you a little bit about what my job involves. One important component is research: I try, together with colleagues or students, to prove new mathematical facts. Once we have proved that something is true, assuming we have not made any mistakes, then it stays true forever. Mathematical proof is eternal. People who like maths often find this certainty very comforting: a maths question has many wrong answers, but it also has a right answer, and if you work hard then maybe you can find the right answer.

In order to achieve this level of certainty, mathematicians need something called “rigour”. My first example of mathematical rigour was one of my maths teachers here at St Margaret's, Mr Douglas Linnett. Every time Mr Linnett wrote an expression on the board, say if he was dividing by x, he would write on the side of the board “x not equal to zero”. I noticed that he did this every single time, and I thought “hmm, that must be important”. (It is! Don’t divide by zero!) Mr Linnett also gave me my first exposure to computer science, as he ran lessons in the lunch hour teaching students how to program in Basic. Today, my research sits at the boundary of an area called discrete mathematics and theoretical computer science.

My research mostly looks at networks, like social networks or transport networks. These networks involve objects, which might be people or cities. Some pairs of objects are joined by an edge, say if the two people in the social network are friends, or if there is a road between two cities. Real-world networks are extremely large, and sometimes there is uncertainty in the information about these networks. So, I study very large networks where the connections are formed randomly: this allows us to think about average cases and try to understand the typical behaviour of these networks. Even though I am happy that my research has real-world applications, it is not the applications that motivate me. I am what we call a “pure mathematician”: I like to work on a problem because it is interesting and challenging, or because the mathematics used to solve the problem is beautiful and fun.

Speaking of fun – teaching is another important part of my job. I really enjoy that moment when I can help a student understand something for the first time. Universities are facing some challenges at the moment, dealing with a decline in student attendance on campus, and the rise in the use of AI. We are having to rethink how we teach, how we assess, and how best to engage our students so that they will want to come to campus. So, it is certainly an interesting time to be an educator. One approach that UNSW has taken, together with many other universities, is to introduce “active tutorials”. Instead of students listening to a tutor explain how to solve a problem, the students work in small groups at a whiteboard and try to solve the problem themselves. This improves their teamwork and communication skills as well as their problem-solving abilities. If you ever find yourself helping a student with a maths problem, I recommend allowing them to play with the material and make their own discoveries. This makes it more likely that they will one day say, “maybe I am a maths person”.

Mathematics is a really good way to train your brain to think logically and analytically. It can also teach us to be sceptical: someone might write down a mathematical statement, but is it really true? I think a healthy dose of scepticism is extremely important, especially with the world as it is right now. So I would encourage all young people to keep thinking for yourself and keep asking questions.

The final component of my job is called “service”, which resonates with the values that St Margaret’s aims to instil in its students. Soon after I started at UNSW, prompted by my Head of School and together with two colleagues, I designed a workshop for young women high school students. The aim of the workshop was to encourage the students to keep studying mathematics and show them the very many possible career paths which can arise from studying mathematics and statistics. This annual workshop, called Do The Maths, is still running now, 20 years later. Our early workshops had around 25 students, and we now have 300 participants each year!

I was also lucky enough to be the Chair of a group called WIMSIG: the Women in Mathematics Special Interest Group of the Australian Mathematical Society. WIMSIG exists to support women, trans- and gender-diverse people working in the mathematical sciences in Australia. Among other things, WIMSIG runs conferences every three years where all the maths talks are given by women, trans- and gender-diverse researchers. When you think that in Australian mathematics departments, on average, less than 30% of the academics are women, you can see that these conferences are very impactful. It is a really powerful thing when women and other underrepresented groups stand together and support each other.

I did my undergraduate studies at The University of Queensland at a time when there were several wonderful senior women mathematicians in the department who were great role models, including Professor Anne Street. Anne was just the third woman in Australia to become a professor of mathematics. By observing her, I concluded that it was possible to be both a leading researcher and a mother. I asked her about this and she said, “It is possible, but it is very, very hard work”. For me, one of the people who makes what I do possible is my husband Martin: thank you Martin for all your support, and thanks very much again to St Margaret's for this award.”