The question: what can Einstein's general theory of relativity do when pushed to extremes? In particular, can it generate naked singularities from smooth initial data?
The tools: simulating extreme spacetimes through numerical time evolutions to discover what happens, then build mathematical models for it (which may also require numerical work).
I am currently trying to verify strong indications from numerical relativity simulations that naked singularities can form from the collapse of very strong gravitational waves in vacuum, at the threshold of collapse. This is interesting because it tests general relativity on its own, without the complications of matter.
I am also interested in gravity in 2+1 spacetime dimensions. In 2+1 dimensions, general relativity is simple (there are no gravitational waves, and the vacuum solution is locally unique), but not too simple (vaccuum black holes exist, and black holes can be formed in the gravitational collapse of matter, if we assume a negative cosmological constant). I am trying to see if and how naked singularities can be generated at the threshold of collapse, as they are in 3+1 and higher dimensions.
Here are some possibilities for PhD projects: