I'm embarking on work involving NLS and Fluid Dynamics under the supervision of Boris Khesin.
Regarding my philosophical outlook, I agree with some of the points made in this article by V I Arnold.
I won't claim it as "my philosophy," (it's too hardcore for me), but This talk given by R Hamming is also p. cool.
I'd like to put together a study group for learning symplectic geometry. The starting point will be the first part of Guillemin and Sternberg's book, Symplectic Techniques in Physics. It begins with a presentation of Gaussian Optics as the prototypical symplectic system. From there it only gets more interesting. There's also a request to learn about normal forms.
- Guillenmin and Sternberg, Symplectic Techniques in Physics
- Spivak, Physics for Mathematicians - Vol. 1
More references for summer learning 2012 will be put up as they are used.