By Gleb Kotousov
Mathematical physics is a wonderful field because the problems that are posed are on the one hand, very mathematical, but on the other, always have an interpretation in the real world. Needless to say that it is mathematical physicists who first described gravity, developed the tools needed to control electricity and thought of countless numbers of “useless” gadgets, such as lasers, which now underpin our modern technological world.
Often, the research done by such physicists revolves around equations called partial differential equations. These are usually notoriously difficult to solve, but they describe the natural world beautifully and it is amazing how many different phenomena a single such equation can account for. For example, the partial differential equation for fluid flows can be used to model traffic, gasses, to understand how aeroplanes fly and to make faster submarines. That is why, despite their difficulty, it is important to study them.
The equation I am working on is important in quantum mechanics, the theory that is used to describe really small things like atoms and electrons, and also has the interpretation of being the distance on a sphere with protruding spikes. Interestingly enough, the techniques I am using to analyse it come from electrostatics. Right now I am trying to solve the equation numerically, using a supercomputer, so that we can understand it better.
Although I doubt that this equation will become as ubiquitous as the famous Navier-Stokes equation, it is still a very interesting problem in its own right and I hope my work will help develop many fields that use partial differential equations.
Gleb Kotousov was one of the recipients of a 2013/14 AMSI Vacation Research Scholarship.