What did you study at A-level and why did you select those subjects?
I studied A Levels in Maths, Further Maths, Physics and Chemistry. I chose Maths and Further Maths because I have always been fascinated by mathematics and enjoy thinking mathematically. I chose Physics and Chemistry because I’m curious about how the world around us works.
What degree did you study?
I completed an integrated master’s (MMath) in Mathematics at the University of Liverpool and am now progressing to a PhD in Pure Mathematics at the University of Manchester.
What inspired you to choose and study your degree subject?
I chose to study mathematics because I wanted to deepen my understanding of the subject. I had always excelled at maths in school and sixth form, and I found it rewarding to learn new concepts. Through popular maths content online, I discovered that there was a vast world of mathematics beyond GCSE and A Level, and I wanted to study it rigorously.
What key skills did you learn at university?
At university, I learned how to formalise my mathematical thinking, approach new areas of research, and identify interesting problems to explore. I also developed my mathematical writing skills and became proficient in using LaTeX to produce well-structured documents.
Additionally, I attended a range of maths seminars, which taught me how to understand cutting-edge research efficiently and decide whether it was relevant to my own work without having to study an entire module on the topic.
What jobs have you had during your career?
I undertook summer research internships between my second, third, and fourth years at the University of Liverpool, where I studied integer geometry. After my fourth year, I worked as the School of Physical Sciences EDI Summer Intern, focusing on improving accessibility in course content, particularly for blind and visually impaired students.
What is your current role and what do you enjoy about it?
I’m currently preparing for my PhD, and although I haven’t officially started, I’ve already begun my research. One of the things I enjoy about being a researcher is having the space to think about a problem that challenges me. I like the sense of accomplishment when I have a breakthrough.
I also appreciate being part of the mathematics research community — sharing my findings with others, learning about their work, and exploring both related and entirely different areas of maths.
Do you have an area of research?
My current research focuses on integer geometry, an area of pure mathematics that uses geometric approaches to study Cartesian products of the integers with itself, which we call the integer lattice.
My research has involved introducing a notion of circles to this geometry. Here, unit circles (or circles of radius 1), have similarities to Farey starbursts and with the set of coprime pairs — a pair (x,y) such that x and y share no common integer factors other than 1.
What has been your most exciting project or career role?
In the final year of my undergraduate degree, I completed a project on the intersection of integer circles. I studied the density of these intersections. The density of an object in integer geometry (or more generally) is, in less precise terms, the proportion of the number of points in the object to number of points in the ambient space (or in our case, Z^2). As Z^2 is infinite, we take limits to find this density.
My research found the density of the intersection of unit integer circles as an Euler product (product over the prime numbers) when we take the limit with respect to larger and larger squares. This sequence of increasing squares is called an exhaustive filtration, and one direction of further research is studying how this density changes with a different choice of filtration.
This project was particularly rewarding because it produced new results and also won a competition run by the Institute of Mathematics and its Applications (IMA), where I presented my findings to a group of mathematicians.
What are your top tips for working in your industry?
If you want to work in mathematical research, my biggest tip is to stay curious. If you’re a student, ask your lecturers and teachers plenty of questions. This helps in two ways:
- Developing better questions – The first questions you ask might be simple, but as you keep questioning, you’ll learn to spot deeper and more interesting problems.
- Building connections – Asking questions helps lecturers notice your potential. Most lecturers are researchers, so they can offer valuable advice and may connect you to opportunities that can shape your career.
Above all, being interested in maths matters more than being naturally “good” at it. Curiosity drives better research.
Another piece of advice: don’t take anything for granted. If you’re told a theorem, try proving it yourself or read the proof carefully. Don’t stop until you’re satisfied that you understand why it’s true. Developing this habit strengthens your foundations.
What is the best piece of advice you have been given?
The best advice I’ve received was to apply for the Martingale Foundation’s PhD scholarship. Another valuable piece of advice came from one of my MMath supervisors, who shared wisdom passed down from his PhD supervisor, Vladimir Arnold: “You should either share your findings with everyone or with no one.”
In today’s context, “sharing with everyone” means uploading your work as a preprint on arXiv (or a similar platform for your field).
Any advice you’d like to share?
Don’t give up on maths. Struggling with a problem doesn’t mean you can’t solve it — it just means you need more time, persistence, or new tools. It’s also okay to take breaks when something feels overwhelming.
Sometimes, a problem remains unsolved because neither you nor the wider community yet has the necessary mathematical tools. Putting a problem aside to work on something else is not the same as giving up.
Also, ask for help when needed. Listen to other mathematicians in seminars, read research papers, and learn from textbooks. Other mathematicians are incredibly insightful — and you can learn a lot from them.
Why are you passionate about maths?
I’m passionate about mathematics because I love the satisfaction of solving complex problems after working on them for a long time.
I enjoy how maths combines creativity with logic, and I take pride in contributing — even in small ways — to our broader understanding of mathematics.
More resources:
Circumscribed Circles in Integer Geometry
Rebecca's LinkedIn