Guidelines for the course paper
The paper is due at the last class. Ideally, the paper need not be long -
4-7 pages are sufficient. It must contain an accurate bibliography. The content
should be physico-mathematical, as opposed to material pertaining to the
history or philosphy of the subject. It should be on a topic which is past
the material of Wald Ch. 1-6 - or in any case something beyond what is taught
in class. One may be very geometric, for example presenting a proof of
a topological property of a spacetime. Or, describe Physics of GR. In most
cases, there should be one clearly written calculation, pertaining to GR.
If it is a tensor calculation, parts that are easy should be written
without indices, if at all feasible. The more involved computational parts
should use indices. It should be clear that the calculation is understood by
the writer (do not just copy it from a book without filling in skipped steps,
at the very least the most pertinent ones.)
Aside from this, the paper must be typeset with introductory notes, explaining
what is shown, giving definitions and the relevant physics, etc. It should also
have a summary, containing relations to other parts of the subject, further
insights, etc. Write these in your own words, not in those of the sources
you are using.
Below are some possible topics. You can suggest to me other ones. In this list,
each topic is a broad subfield. Within the one you choose, select a subtopic
which fits the size requirements above. The associated work need not start
from first premises, as long as the assumptions are clearly presented, and what
you write from there on is fairly complete and self-contained.

Suggested topics
Topology of Lorentz manifolds
Geometry of gauge Theory
Spinors
Complexified Spacetime (twistors)
Variational approaches to General Relativity
Gravitational waves
Gravitational lenses
Rotating black holes
Hawking Radiation
Superstars
Energy in General Relativity
Cauchy hypersurfaces (Initial Value formulations)

There are many mathematical-writing software programs. You can
use any of them. A common one is LaTex/Tex. Links to various versions
of it can be found on Professor Pugh's web page.

Online versions of the course papers our available here.