Geometry in dimensions higher than 3 may seem to be a mathematical curiosity at first, but actually abstract geometric methods have proven to be indispensable when attacking various physically motivated problems. For instance, the geometric properties of the space of all possible configurations of a physical system can tell a lot about the behavior of the system. In this project, we will focus our attention on a specific family of systems which is relatively simple but displays interesting behaviors. The system has several parameters and we will study how the behavior of the system can suddenly undergo certain qualitative changes while varying the parameters. This will entail studying the system from a pure math point-of-view, and also developing computer programs to run simulations of the dynamics of the system and producing useful visualizations of the system in motion.
Vector calculus and linear algebra. Experience with higher math (especially algebra or classical mechanics) would be helpful, but is not strictly necessary.
We will be doing some coding, but the language is up to the preferences of the team. Python and mathematica are the most likely to be used.