By Alastair Dyer
Loosely speaking, we can think of covering a sphere by taking the surface of an n-torus (the surface of the shape given by gluing n doughnuts together) which is made out of elastic, and stretching it in such a way that it covers the surface of the sphere. Unfortunately this is almost always impossible to visualise as we also require our elastic to be magic and able to pass through itself!
When we formalise this idea in mathematics we find that there will be certain special points in the covering called branch points where the elastic twists around itself while leaving a single point fixed. If we specify the nature of these branch points we can then count the number of ways to cover a sphere. This is known as a Hurwitz number and counting these coverings is a fascinating problem which combines geometry and algebra, and has important applications in theoretical physics and string theory.
An important theme in mathematics is to take an abstract problem and reduce it to something concrete. In this case we can simplify our problem by converting it into algebra using a fancy tool called the Infinite Wedge Space, which is useful for many combinatorial problems. Here it allows us to calculate Hurwitz numbers purely in terms of algebra which is much easier.
Taking part in a pure maths research project was both a fun and challenging experience. Trying to find a new proof is a much more exciting and difficult exercise than what I was used to from my undergrad courses and I look forward to continuing in maths research in my honours year.
Alastair Dyer was one of the recipients of a 2013/14 AMSI Vacation Research Scholarship.