Kyan Jenkins
The University of Adelaide
I am commencing my Honours in Mathematical Sciences at the University of Adelaide in 2025. My thesis is concerned with proving that three commutative squares important to fixed point theory are quasi-categorical pullbacks; this is with respect to the proof of a slightly weaker statement by Michael Shulman. I am interested in all things algebra and topology, and enjoy the critical thinking, patience and creativity that pure mathematics offers. Outside of my mathematics study, I work as a cook in multiple restaurants and coach high school debating.
Can you give me a quick rundown about the type of mathematics you are studying and its potential impacts for the broader community (think how you would explain your work and studies to others who don’t study maths)
Category theory is a relatively new field of mathematics which began as a tool to assist topologists in the 1940s, originally only to serve a niche within the field of algebraic topology. A mathematical concept is a category if it can be described in terms of objects and maps between those objects satisfying basic requirements like identities and associativity (for example sets and set functions, topological spaces and homeomorphisms, etc.). The rapid development of category theory revealed that it is useful to not just algebraic topology, but almost any mathematical idea or gadget which can be understood as a category. In particular, the foundational brilliance of category theory as a study is that the most important properties of categories are determined almost entirely by maps between the objects instead of the objects themselves.
In fact, using this observation as a driving force, we can think about maps between maps in categories, and then the categories, called 2-categories, which instead of having just objects and maps, have objects, maps and maps between maps. Taking this further, there are categories, called 3-categories with objects, maps, maps between maps, and maps between these maps between maps. This pattern continues, but each increment does add increasingly overwhelming complexity as it becomes increasingly hard to write down a coherent and rigorous definition for such an ”n-category”.
The theory of 2-categories (concerning examples like this), is already quite convoluted, and the theory of 3-categories even more so. Surprisingly then, higher category theory (specifically, the study of infinity-categories, where there is an infinite iteration of these maps between maps) is arguably more well understood than even the theory of plain old 4-categories. This is in part due to the large variety of definitions/approaches that one can use to model an infinity-category, which all turn out to be consistent with each other. My honours thesis is about giving an alternative proof of a theorem in higher category theory using a model which is more popular in the literature than the original proof.
How did you get into mathematics/statistics/data science? Was there someone or something that inspired you to this field?
My excellent year 8 mathematics teacher helped me realise that maths is not about just solving problems but is instead about learning how to formalise and solve very abstract problems. This got me interested in mathematics, though even up to year 12 I was considering doing engineering or physics. My specialist mathematics teacher in year 11 and 12 helped me realise that my passion lied in the abstract thinking of (pure) mathematics, and since then I have never looked back.
You received a Travel Grant to attend AMSI Summer School 2025. How important was this in terms of your ability to attend, fully participate in the program and meet others studying in similar fields? Do you think it was an advantage to attend the program in-person?
I certainly think that attending in person was of great benefit; it allowed me to engage to the highest degree in the course, being able to discuss problems and continue learning outside the tuition hours. Thus, I am very grateful for receiving the travel grant. Without such a grant, it likely would not have been financially viable for me to attend, and I either would have completed the course remotely (and likely struggled) or not been able to attend in any capacity at all.
What was the most valuable part of the program for you?
The most valuable part of the program for me was being immersed in a mathematical culture external to my own university. It was great to meet other students, postgraduates and researchers in mathematics, as well as to feel like a student at a new campus. It has convinced me that there is a whole world of mathematics to be explored, and a great network of mathematicians to meet, learn from and collaborate with in the future.
In the long-term, what do you think are the benefits of having attended Summer School?
First and foremost, the class I participated in at AMSI Summer School developed very rapidly my maturity as a mathematician. As my first honours-level class, and in such an intensive time, it has been an excellent introduction to the taught versus self-taught balance I need to adapt to. The other immediate benefit is the great network I have been able to interact with. Attending the Summer School forces you to form friendships with other mathematicians who likely share your research interests to some degree and also provides the ability to meet current lecturers and researchers who are well established in their field. The networking opportunities do not stop there though. For example, I attended the Australian Category Seminar at Macquarie University, which was an excellent opportunity for me to meet researchers and students in fields very relevant to my own, providing clear medium-term benefits.
Summer School included a special Careers Day program which aims to help give students an idea of the kinds of career paths available to maths graduates in industry and private sector research areas. Do you feel better equipped to explore career options in the mathematical sciences after attending AMSI Summer School?
The careers day program was very useful in alerting students like myself to the opportunities in the corporate world. For example, by advertising graduate programs and internships, it has made me aware of deadlines and opportunities both local and international, as well as providing insight into day-to-day life in jobs like these.
What advice would you give to someone who is considering applying for Summer School in 2026? Should they apply and why?
I would strongly recommend attending Summer School if you are interested and able. It is an excellent social and academic opportunity, which has surely made me a better mathematician in many regards.
What are your current career ambitions in the mathematical sciences sector?
I am currently hoping to pursue research by a higher degree, whether it be a masters or a PhD after I complete my honours.
How did connecting with the community at AMSI Summer School support your experience?
It was very exciting to be immersed in a community of excellent mathematical students and researchers, and certainly very motivating. It was a very good and rare opportunity to make connections which will surely pay off long term as a relatively early point in my mathematical career.
Any other feedback/comments you would like to provide on the AMSI Travel Grant or AMSI Summer School 2025?
Hold yourself to as systematic and repeatable routine in terms of study. I found the content much too dense to leave till the last minute and would not have enjoyed myself had I done so. Developing this routine is also the only way to allow yourself to make the most of the social experience, so that you can enjoy yourself when you are not studying.