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Are there any open problems in physics involving Lie groups and differential equations for a phd theses.

Some applications are say, Noether's theorem in classical or quantum field theory. But I am not sure if those topics lead to any research problems.

So any idea about prospective research problems in application of Lie groups to differential equations?

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I do not believe that there are any.
You can check Stephanie Singer's book on Lie groups as applied as symmetries of differential equations, and also the book on mechanics and look at the unfortunately old-fashioned review http://people.ucsc.edu/~rmont/papers/Symm_in_Mech_Review.PDF

There you will see that although there is some research activity, it is what a physicist would consider "pure math", e.g., the possibility of collisions in the three body problem. And this activity, also in Symplectic Geometry and Lorentzian manifolds, takes place in the mathematics community, it is not done or interesting to physicists. And Prof. Singer herself is now doing Statistics...just like me: she left Lie Groups to do Stats, as I did too.

Statistical Mechanics is the future of Physics.

Now Lie Groups do play an important role in Statistical Mechanics, see Mackey's wonderful review article in the Bulletin of the American Mathematical Society, and Volume 4 of Gelfand and Naimark's Les Distributions: applications de l'analyse harmonique (I am sure there is an English translation, too). And some of that activity is indeed Physics, although what is grouped around Ergodic Theory and particularly interested Mackey was purely mathematical.

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    $\begingroup$ "Statistical Mechanics is the future of Physics." This is quite a bold statement, could you elaborate on it what you mean with this? $\endgroup$ – Hunter Jun 1 '14 at 16:17
  • $\begingroup$ It is so bold that it would require an entire survey talk to answer your comment, and I'm not qualified to defend or prove the statement anyway. But: the theoretical concepts are exciting, require new maths, and one can get the experimental evidence that is relevant. In contrast, cosmology and fundamental theoretical particle physics are at a dead end in that regard. I am a string theory skeptic, as are many physicists, in private. The whole string theory community is a case study in what happens when you have no experimental evidence. $\endgroup$ – joseph f. johnson Jun 1 '14 at 16:24

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