## Please suggest some good topics for Maths research paper

There is a student in US. He has completed 4 hours of high school and is applying for college. As part of the application procedure, he has to submit a research paper in Mathematics. He is a very good student.
He accelerated math and enrolled for Differential Equation, while still being in high school, although differential equations is a college level course.
The colleges to which he is applying want to see a Math Research Paper.

I am not from US. So I do not know what kind of paper will be good. Could you suggest some good topics?
And also, any idea how long should the paper be (i.e. how many pages?)
Please give any other details you think appropriate.

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 He could investigate one of the millenial math problems not with the idea of solving it but in understanding and describing to others. http://en.wikipedia.org/wiki/Millennium_Prize_Problems I think they are looking for some depth from your friend something that he has looked into beyond what is taught at school. Another topic would be mathematical origami which has to do with computational geometry; http://en.wikipedia.org/wiki/Mathema..._paper_folding
 Than you. I am looking for some topics in differential equations. The research paper has to be submitted by only those students who accelerated their Maths course. This student accelerated and took differential equations. So I think the colleges expect the paper to be related to that.

## Please suggest some good topics for Maths research paper

He has studied the following:-
* vectors in Euclidean space
* vector analysis
* analytic geometry of three dimensions
* curves in space
* partial derivatives
* optimization techniques
* multiple integrals
* vector fields
* Green's theorem
* Divergence theorem
* Stokes' theorem

So, I am wondering what interesting topic to choose which involves one or more of the above.

 okay navier-stokes equation for fluid flow its a millenial problem quite difficult and quite interesting as well.