How can we study a four dimensional space? There are many tricks for this, and very useful trick is to exploit some symmetries of the space to reduce the dimension of the problem. In this project, we will study physically motivated questions about four dimensional spaces by instead looking at 2 dimensional polygons which hold most of the information about the higher dimensional space. The polygons themselves can be complicated, but it is much more concrete than working with a four dimensional space. In particular, we will compute certain interesting invariants of the four dimensional spaces by solving an optimization problem on the associated polygon. There are different types of polygons, and this problem has essentially been solved for the simplest polygons (called Delzant or toric polygons), but there is a more complicated case (called semitoric) which remains open. The semitoric case will be the focus of this project and, time permitting, we may also explore the question in even higher dimensions.
Vector calculus and linear algebra. Abstract algebra would be very useful, but not completely necessary.
The project will use a python program we wrote previously, so being familiar with python would be useful. That being said, there are many components to the project which are not directly related to writing code, so anyone interested in the project who has no experience with python (or programing in general) is also welcome.