Tiernan Cartwright
The University of Sydney
Tiernan Cartwright is a third-year PhD student at the University of Sydney (USyd), supervised by Zhou Zhang. He currently studies the complex Monge-Ampère equation and the Kähler-Ricci flow. These are two important equations in Kähler geometry, a mathematical area which he learnt about during honours (which he completed in 2023, also at USyd).
Give me a quick overview of the type of mathematics you are studying, and/or the aims of your research and its potential applications/outcomes (how you would explain your work and studies to friends who don’t study maths?)
For a general audience: Differential geometry is the study of the curvature of smooth surfaces, such as spheres, donuts, or saddles, to name only a few. It is the maths needed to describe gravity. In Kähler geometry, this smooth structure interplays with a complex structure, which is a geometry coming from the complex numbers (involving the imaginary number i, introduced to act as a square root of -1). I study this from a purely mathematical perspective, with the goal of better understanding possible geometries. However, in some models of string theory in theoretical physics, Kähler geometry is speculated to be related to extra dimensions of the universe.
For mathematicians: I study partial differential equations (PDEs) that arise in Kähler geometry. Apart from PDE and geometric techniques, I also use techniques from several complex variables, specifically pluripotential theory, which is a complex variables analogue of potential theory, the theory of subharmonic functions.
How did you get into the mathematical sciences? Was there someone or something that inspired you to this field?
I’ve always liked maths. However, a particularly formative moment was taking my first semester of maths at university. It was challenging and a big step up from what I was comfortable with at high school. Yet it was engaging and I began to realise that maths is a much wider area than I’d previously thought – something I am still coming to terms with today. There is so much to learn.
Winter School is designed to give students a deeper understanding of their area of research and expose them to others working in different fields/industries. Tell me about your Winter School experience. What was the most valuable part of the program for you?
The most valuable part was the lectures: getting a chance to hear from leading experts about recent developments in and perspectives about geometric analysis, the subject of the 2026 Winter School. A close second was having the chance to communicate my research, both formally during the participant talk session, and through informal discussions during the break times.
What was your main take away/s from AMSI Winter School? Something you learnt? A connection you made? Do you have new ideas for your work/research or see it in a new light?
I particularly liked learning about Einstein metrics in the two first-week courses. This topic is relevant for my research, but one that I’ve not had the chance to study much yet. In the first course, I learned about how to use symmetry assumptions to study examples. The second course focused on dimension 4 and connections to physics. The ideas were very interesting and are good to be aware of. If I continue in the same research area, I’m sure I will use several of these ideas eventually.
What surprised you most about Winter School?
Seeing how each lecturer attentively listened to the other lectures during their week, following along and asking questions. This gave me the impression that the topics were more connected than I thought, both on a mathematical level and in terms of the people who study them.
What is the importance of in-person communication at an event like Winter School?
It is vital for getting to know the other participants and networking, as you end up having many more conversations compared to attending online. It is also very useful for learning the content: in-person discussions allow you to consolidate your understanding, ask more questions, and it’s more motivating.
You received a grant to attend AMSI Winter School. How important was this in terms of your ability to attend, fully participate in the program and meet others studying in similar fields?
I wouldn’t have attended without the travel grant: I would have saved my limited PhD travel money for other occasions. At least, I would not have attended in person, which was very important for fully participating in the program and meeting people, as mentioned in the previous question.
What advice would you give to someone who is considering applying for Winter School? How would you describe the conference to them?
If the theme of the Winter School is at least somewhat aligned with your research area, you should apply. The Winter School is a chance to study advanced topics alongside a community of fellow students and researchers: it is currently the only conference of this kind in Australia.
Tell us about a moment that changed a preconceived idea about your field?
The lecturers distilled complex topics and presented a clear picture. This affirmed my confidence in my ability to do research in the field. Although I know mentally that it’s possible for me to contribute, sometimes it’s hard to feel confident due to the difficulty of the area.
Where do you want the mathematical sciences to take you? Where do you see yourself in five, ten years time?
I would like to become a research mathematician, so hopefully by then I will be employed as a lecturer at a university. I am also open to working in industry, but it must be in a scientific role: I like science as it’s interesting while also having a positive impact on society.
Any other feedback/comments you would like to provide on the travel grant or AMSI Winter School?
The conference venue at the University of Queensland and accommodation at the Women’s College were delightful. I thank AMSI for facilitating the Winter School and offering me a travel grant.