Interesting topics to research in mathematical physics for undergraduates I'm planning on getting into research in mathematical physics and was wondering about interesting topics I can get into and possibly make some progress on. 
I'm particularity fond of abstract algebra and topology and if possible any topics that involve abstract algebra would/topology/calculus of variations would be especially appreciated 
 A: I'm guessing you're a final year maths/physics degree student ?  In your comment you said you'd prefer to have an original research component in the paper.  That's quite tough since when people do the grad school, they tend to spend a year or two learning the extra specialist material they need to get them to the point where they can contribute to original work in a field.
Realistically, if you don't have a vast amount of time to learn new stuff yourself, then what you can achieve is constrained by the knowledge level of your mentor.  I would recommend that developing new cutting edge stuff is too hard, so a more realistic target would be to do some "original" work by applying some existing advanced techniques to a particular problem.
As I said, it depends on your mentor, but an example of sort of thing I mean would be to take a specific classical mechanical system (something with constraints to make it nontrivial), and describe it geometrically.  Set up, in differential geometric terms, the phase space, Hamilton Jacobi equation etc.  If you have time, see if you can apply the techniques of geometric quantization to it.  This would fit in with the particular fields you mention.
A: This doesn't match your stated interests, but the things you talk about are generally well established fields with an enormous history, and it generally takes longer to get to the forefront of research in such areas.
If you can code, there is the interesting problem of classical turbulence. It is wide open as mathematics and as physics, and there is a potential for much progress with simple numerical schemes, and ad-hoc  ansätze for the steady state.
Consider a 2+1 dimensional nonlinear field theory. You can use any field theory, but stick to 2+1 dimensions because in 1+1 dimensions, you tend to hit too many integrable systems when randomly futzing around. A good model system is Schrodinger field turbulence
$$ i{d\psi\over dt} = -{\nabla^2}\psi + \lambda |\psi|^2 \psi + f(x,t)$$
Then set up a long-wavelength source for the field, vary $\psi$ using $f(x,t) = C(t)$ where $C(t)$ is some random oscillating function.
There is a general fact that these types of equations will move the energy from long wavelengths to short ones, reflecting the fact that energy (and "particle number", total $|\psi|^2$) is conserved. There are many more short-wavelength modes than long-wavelength ones, so you generally get a cascade which moves energy from long-wavelength to shorter ones in a scale invariant way.
The limiting statistics of the scale invariant distribution is not known for most nonlinear equations, not even numerically. For the Navier Stokes equation in 3+1 dimension, describing this cascade is the famous problem of fluid turbulence. But turbulence is complicated by the fact that the mixing time for each mode is entirely determined by the nonlinearity, because the viscous term is irrelevant in the mixing regime. For the equation above, you have frequencies for modes which should be comparable to the nonlinear mixing, and there are going to be interesting phenomena, none of which are investigated in any sort of depth.
This problem is easily approached with cookbook numerical methods, but the statistical description is essentially as deep as you like. The articles of Kraichnan regarding fluid turbulence are important, but you don't have to know anything to work this stuff out.
This problem is known as superfluid turbulence, and it might be too well studied. But if you consider other nonlinear systems, you can easily find systems which have never been considered at all. The mathematical problem is well defined for nearly any nonintegrable nonlinear partial differential equation with a conserved energy. If you stick to 2+1 (or 3+1 or more) dimensions, this is essentially every example. To be specific, consider surface water waves driven by a steady wind. What are the statistics of the wave-crests?
I think this is the most important problem in PDE's today, and it is completely neglected because of the paucity of new ideas and interesting examples.
A: Check this list of courses offered. other universities also would be on a similar lines, but this one atleast would give you an idea.
A: Dijkgraaf Witten theory calculations with group cohomology are great for abstract algebra and topology. Lots of examples someone in your position can compute. If you want a specific one, go ahead and ask.
