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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

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do you mean original research? Or fun beyond-class-material area? IMHO, probably the best thing you can have as an undergraduate wanting to do research is a great mentor--so maybe you want to talk to professors you know and like about what you could do that they'd be willing to guide you in. – Jamie Banks Feb 1 '12 at 0:54
I guess original research in some sense. I want to be able to potentially write a modest paper on it. I do have a great mentor. He has suggested that I think about and come up a few ideas which I'm interested in first. – Rebel Feb 1 '12 at 1:34
Cross-posted to TP.SE – Qmechanic Feb 4 '12 at 16:59

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 ansatzes 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.

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I should add that I am not sure about the dynamics of the equation I gave, but I could list nonlinear nonintegrable equations all day, and the turbulent cascade in each of them would be an open problem which is not investigated. – Ron Maimon Feb 2 '12 at 14:07

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.

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can you give me some specific examples of classical mechanical systems which I can model mathematically and a very brief outline of how I'd go about doing that? – Rebel Feb 2 '12 at 13:30
I was thinking of something along the lines of, say, a particle constrained to move in a circle - a bead on a circular wire if you like. The configuration space is a circle. Adding the possibility to specify a speed at each point, your configuration space gets itself a tangent bundle - a differentiable manifold on which you can specify a Lagrangian. Add a potential to make things interesting. You can then define a phase space (cotangent bundle), perform the Legendre transform, and look at the Hamiltonian flow. With a bit of software you could probably plot the flow. cont... – twistor59 Feb 2 '12 at 13:53
cont... If you learn a bit about geometric quantization you can define Hermitian line bundles which describe the space of quantum states for your bead on a wire. The reason I suggested this was that it cuts across most of the topics you mentioned you were interested in. But your mentor is the one who needs to have the final say! – twistor59 Feb 2 '12 at 13:54

Check this list of courses offered. other universities also would be on a similar lines, but this one atleast would give you an idea.

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I was just looking at this. Could you maybe give me more specific project ideas that I can look into rather than broad topics? – Rebel Feb 1 '12 at 2:17
I am not a physics major, so I do not have much idea regarding this, just studied the subject as a matter of interest so got some idea of it.. thats where it ends – Jack Feb 1 '12 at 2:51

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