Max Jolley is currently studying as an undergraduate student at Monash University, having completed the International Baccalaureate in 2012. In January 2012 and 2013 he attended the National Mathematics Summer School at the Australian National University, where he was introduced to and developed an interest in many areas of pure mathematics. He has just completed the second year of a bachelor of science with a double a major in pure mathematics. So far he has studied calculus, analysis, differential geometry, algebra, and number theory, along with undergraduate physics. He is planning on completing two research projects in the next few months: one into algebraic geometry over the summer and one into Galois Theory next semester, and taking an honours year in 2016. Outside of academia, Max is interested in music, and enjoys playing piano and guitar alone and in a group.
Counting coverings of the sphere
Suppose that we have two surfaces and wish to wrap one around the other. Mathematically speaking, we would like to consider functions from the points of one surface to the points of the other. Of course, a sphere can wrap once around another sphere in a nice and simple way. However, if we use more complicated surfaces or try to wrap many times, then there are necessarily points where the function is no longer smooth. By controlling the location and behaviour at these so-called branch points, one can try to count the number of such coverings. The project aims to prove a conjecture concerning the structure of this enumerative problem in the so-called spin case. We will approach the problem using linear algebra, group theory, complex analysis, and ideas from mathematical physics.