Ayesha Bennett is a first year PhD student in the Department of Pure Mathematics and Mathematical Statistics, supervised by Professor Holly Krieger. She tells us about the fascination of fractals, what it's like to study at Cambridge, and Cambridge Blues football.
I've always been fascinated by fractals, even before I fully understood what they were. Most of the objects we meet in school geometry have dimensions that are positive integers: for example, the line is a one-dimensional object and the plane is a two-dimensional object. Fractals, by contrast, are objects whose dimension is not an integer - they live in-between dimensions. The idea that an object could have a dimension outside of a positive integer seemed both strange and exciting, especially since these kinds of patterns also appear in nature, for example in coastlines or ferns.
However for me, mathematics doesn't need to be grounded in the "real-world" to be thrilling. Complex dynamics (more on this below) explores the properties of famous fractals like the Mandelbrot set, which are both visually and mathematically beautiful. The study of such fractals reveals deep patterns, both in the behaviour of dynamical systems that are related to particular fractals and in the parameter spaces that describe entire families of such systems. To understand these patterns, we rely on tools from many areas of mathematics - number theory, ergodic theory, arithmetic dynamics to name just a few. Seeing the links between all of these areas is one of the aspects of study I find most compelling.
Complex dynamics is the focus of my PhD. A dynamical system is something that changes over time. Examples from the real world are the weather or the growth of a population of animals. In mathematics dynamical systems are typically generated by repeated application of a mathematical function. If we pick an initial point, we can observe its trajectory through the underlying space. In complex dynamics, we look at systems generated by complex functions, which send each point on the (complex) plane to another point on the plane.
Typically (for example when the functions are complex analytic) the process divides the plane into two disjoint sets. Initial points that lie in the Fatou set have trajectories that behave quite predictably. On the Julia set, by contrast, the system exhibits chaotic behaviour. Julia sets are typically fractals — understanding the geometry of these fractal sets is a fundamental part of complex dynamics.
In my research I use measure theory to capture information about the evolution process of certain complex dynamical systems, which naturally arise in number theory. This approach can give information, for example, about the frequency of digits in the base expansion of real numbers.
One of my favourite mathematical moments happened during my year abroad at the University of British Columbia (UBC). I was desperately trying to finish off some knot theory homework on a Friday night before going out with my (non maths) flatmates. They saw me struggling, sourced some shoelaces and joined me at the kitchen table to try and help me figure out the Jones polynomial of a certain knot. Nothing like a looming deadline to get things done!
I met some extraordinary people and learned some fantastic maths during my time at UBC. I made lifelong friendships from working and being stuck on a question for hours, and then doing something social afterwards. It was my first time being in small, discussion-based classes, where we were assessed through presentations and problem sheets instead of traditional exams. I learned far more through this collaborative style of teaching than studying for an exam could offer.
Having studied in London for my undergrad, moving to Cambridge for my PhD was like entering an entirely new world. There are beautiful buildings, a winding river, cows (!!), grass tennis courts, riverside pubs and everything is within an easy, and extremely flat, bike ride away. The environment is ideal for research: I’m surrounded by curious, smart and motivated people from all disciplines. The PhD students, post docs and faculty all share the same building, and there is a huge number of research seminars and reading grounds across all areas of maths, as well as a steady flow of external speakers and guests. This all has made for a great first year of my PhD. However, I do miss the diversity, culture and life of a big city like London! It is sometimes nice to escape from the university environment and remind yourself that life does exist outside of it, which can be tricky in Cambridge.
Outside my studies (and sometimes maybe a bit too much "inside" my study day too!) I enjoy being outdoors, and playing sports. In particular, I have played football since I was seven, and have been playing for the Blues team here, which has been a lot of fun. I have always found it a great outlet from my work. I also enjoy reading—one of my favourite authors at the moment is Benjamin Myers, whose novels are often set the Northern English landscape which I grew up in and love.