There is a convincing and prevalent rhetoric that beneath mathematics lies a deep and elegant mystery, and that the questions to ask arise naturally in the pursuit of uncovering the figurative mystery. To some extent, I find this to be true, and if it were not, I doubt I would be studying mathematics at all. But I find recently that questions like “why does this abstract object behave this way?” get replaced by questions like “what abstract object do I even want to study?”. From further discussion with the researchers in my immediate vicinity, it becomes clear that a large part of Research™ is dealing with the latter questions.
I was prompted to reflect in this way after spending a short time working in private industry. I had previously studied stochastic PDE with a lecturer who ensured I had all the tools to solve complex theoretical problems. However, it was unclear how to formulate these problems, and what one would expect the answer to be. After spending a few months working daily with options, I suddenly had a list of questions to ask about the correct way to price options. I initially believed that an academic would probably not be interested in these questions, which usually pertained to extremely ad-hoc strategies to turn a profit. But my lecturer felt strongly that these ideas would lead to solid theoretical foundations for studying not only options pricing, but SPDE as a whole.
I began to then reflect about most of the mathematics I’ve learnt during my time at university. The process goes like this:
- Be presented with a physical or non-mathematical problem
- Formulate a mathematical statement representing this problem
- Solve this problem with the physical intuition as a guiding hand
- Solve newly created problems with the same intuition
I felt that the development of SPDE paradigms were heavily informed by options pricing principles. The development of parabolic PDE theory was heavily informed by physical intuition of heat flow. The development of Riemannian geometry was heavily informed by our physical understanding of spacetime curvature.
I write this blog post to hopefully convince more future mathematicians to spend time researching and studying adjacent fields of research. A large part of our future research journeys will involve coming up with questions that we hope have reasonable answers. If the world around us already shows us the answers, we can come up with the questions.
Lekh Bhatia
Australian National University
