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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

My mentor is involved in work dealing with topological field theories in regards to developments in string theory.

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I think it's better to have a problem in mind rather than the techniques you want to use to solve it. You might not know which techniques should be used ahead of time! –  Kernel Feb 2 '12 at 18:59

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You might want to try to understand what electric and magnetic charges are, and the integrality constraints that they obey. This would not be research, but it involves nice algebraic and differential topology, and would make a good topic for independent study.

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I spent several happy years in entanglement theory during my BSc and MSc days. I was initially interested in the monogamy of entanglement, which was first discussed by Bill Wootters and what I think were two undergraduates -- see http://arxiv.org/abs/quant-ph/9907047. Tests for entanglement include the Peres-Horodecki criterion http://en.wikipedia.org/wiki/Peres%E2%80%93Horodecki_criterion, which is proved by direct application of the Hahn-Banach Theorem (a foundational theorem for Functional Analysis). Three qubits (two-level quantum systems) can be entangled in two completely inequivalent ways: GHZ and W types http://pra.aps.org/abstract/PRA/v62/i6/e062314. These results are all accessible to those with only an undergraduate knowledge in mathematics and quantum physics. The reason is that entanglement is rather amenable to study via the sort of techniques you seem to favour. Entanglement is defined as a lack of separability, which can be tested for in all sorts of sexy ways (the Peres-Horodeki criterion is pretty mouthwatering, and led to the more general idea of developing "entanglement witnesses"). These kinds of buzzwords might be interesting to both you and your mentor.

Good luck!

Edit: good review article for all things entanglement -- http://rmp.aps.org/abstract/RMP/v81/i2/p865_1

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A tangential point - couldn't help noticing that your linked RMP paper has four authors all with the same last name! Is that pure coincidence or is there some systematics to it? –  user6818 Feb 4 '12 at 4:28
Anibit: It's my understanding that they are all family, for example: en.wikipedia.org/wiki/Pawe%C5%82_Horodecki –  Noon Silk Feb 5 '12 at 0:48
I'm sure it's just a co-incidence... ;-) –  Kernel Feb 5 '12 at 4:18

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