We are all familiar with how mathematics is useful for describing the physical world, but there is another relationship between these two disciplines that is less well known to the general public: physics can also be very useful for investigating purely mathematical problems – more specifically, geometry.
Imagine a liquid flowing through a tube. It’s clear that the geometry of the pipe influences how this happens. This is a clear example of how mathematics (the geometry of the tube) influences physics (the way the liquid flows), but if the geometry of the tube is unknown, we can try to do the opposite, i.e. observe how a fluid flows to try to infer the geometry of the tube. This is an example of how Physics can be useful in illuminating our understanding of a mathematical object – in this case, the geometry of the tube. The main idea behind my research is not much more complex than what is made clear by this analogy. I use certain equations inspired by physics, modeling a kind of non-linear electromagnetic field, in the hope of understanding which physical properties of these fields are purely the result of the geometry in which they propagate.