I made a wiki page, which is mostly a copy of last years page, here:  
http://wiki.sympy.org/wiki/GSoC2010Ideas . I'll try to clean it up a bit.

Any polynomial algorithm will require a good understanding of abstract algebra. 
 Actually, I think most computer algebra algorithms, like the Risch algorithm, 
have their theoretical roots in abstract algebra.  There are also algorithms 
for pure abstract algebra, like manipulating groups (see the GAP project).  But 
I am not highly familiar with them, so I am hoping Mateusz or someone else will 
be able to advise me.

Aaron Meurer
On Jan 27, 2010, at 3:12 PM, Ondrej Certik wrote:

> On Wed, Jan 27, 2010 at 1:39 PM, Aaron S. Meurer <[email protected]> wrote:
>> I would be more interesting to me to learn some cool algorithm and implement 
>> it, though I don't know anything about Quantum Mechanics.  :)
>> 
>> I could also very easily extend my project from last year and spend the 
>> summer implementing ODE solvers.  There are enough methods not implemented 
>> to fill several summers, but I would have to learn them on my own, since I 
>> haven't taken any ODE courses since last Spring and I already implement most 
>> of what I learned there.
>> 
>> But, ideally, I would like to apply what I have learned in Abstract Algebra, 
>> and also it would be better to go this way because most people have not 
>> taken such a course.  Also, abstract algebra is an area of mathematics that 
>> I find particularly interesting, so learning some cool algorithms that apply 
>> it would be the most interesting to me.
> 
> So what exactly? Does it have any applications?
> 
>> 
>> I probably am not the one to do it, but I think we should try to get someone 
>> to do the cython core for a project.
> 
> Definitely. We should start a wiki and add interesting projects in there.
> 
> Ondrej
> 
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