This isn't my proposal, but I do think there is a lot that can be done here.
There are a great deal of tools and results in Algebraic Geometry that can 
be implemented for 3d (and n-d of course). I recommend THIS 
BOOK<http://books.google.com/books/about/Elementary_Differential_Geometry.html?id=UXPyquQaO6EC>
 as 
a good introduction to the subject with a lot of results. Reduction of 
quadric surfaces is a good example as is finding the fundamental forms of 
surfaces.

On Sunday, February 23, 2014 7:08:56 PM UTC-8, Aaron Meurer wrote:
>
> I think it's a good idea, though you'd need to convince me that 
> there's enough work in it for a whole summer. 
>
> Aaron Meurer 
>
> On Sun, Feb 23, 2014 at 3:39 PM, Akshay <akshayn...@gmail.com<javascript:>> 
> wrote: 
> > @asmeurer Could you post your reviews on this one? 
> > 
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