What does the world actually look like if one is travelling close to the speed of light? There are several relativistic flight simulators and animations on the web, eg look here
There are several strange effects, collectively called relativistic beaming:
Another remarkable property is that spheres appear as spheres, they are not flattened as you would expect due to Lorentz contraction (see here). The reason for this difference is that an observer does not "see" his/her reference frame, but rather the photons reaching his eyes, and the photons from the back of the sphere, having further to travel, left slightly earlier than from the front, counteracting the effects of the Lorentz contraction.
Mathematically, the fact that spheres appear as spheres property is related to the concept of conformal (or more precisely Moebius) transformations (see here and refs therein.) The conformal group has many applications throughout physics and mathematics. For example, insights along these lines led Penrose directly to the concept of twistors, useful tools in physics.
In this project we will begin by exploring what a relativistic observer actually sees and the mathematics behind this. Extensions of this idea could then go towards including general relativity, ie examining the mathematics of a black hole and what an observer would actually see near a black hole (see this), or alternatively focusing on the mathematics of the conformal group and its various applications in physics eg twistors etc. Alternatively you may want to explore the related concepts of projective geometry and its relation to perspective drawing. A student might also wish to try to write his own simulations.
Special Relativity & Electromagnetism II.
Search the web, including the links above, there is lots out there. For the mathematics you could start with the following.