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I hope John won't get angry with me for starting yet another blog post project without finishing the others first, but I just need to get some of the stuff out of the way by making a brain dump on some Wiki pages, and a blog stub seems to be a good start:
This will try to explain how fluid flow can be modelled by diffeomorphisms, how certain diffeomorphism groups can be seen as infinite dimensional Riemannian manifolds and how certain nonlinear partial differential equations arise as geodesic equations. This is just too cute to go unmentioned. Related pages are Analytical hydrodynamics and Burgers equation.
Maybe this will grow into more than one blog post. The end should explain that the Burgers equation describes breaking waves, with the breaking of the wave being a singularity of the solution, and what the paper
says about this (which I don't yet understand).