Independent High School Research StackExchange Gurus!
I'll keep this question short and to the point.
I'm going into my senior year of high school, will have an independent research period, and have few ideas of what to do during this period.
Right now, I was thinking about doing analysis of large data sets (see "Super Crunchers" book). However, I can't faind any large data sets to use and, if I did, I wouldn't know where to start.
However, I'm open to ANY ideas for projects. I would like to stay near the engineering/phsyics realm if it were to be data set analysis, since those are my most likely college majors, alongside math. I only know up to calc at the moment, but am willing to teach myself more. Currently, my favorite topic is fractals.
I can program in Mathematica if that helps at all.
Thank you so much for ANY tip you can provde me.
If you have any questions, feel free to ask them.
PS Anyone who wants to be my mentor is welcome haha :b
 A: Ok, you asked for 'anything' :)  How about using mathematica to display representations of Lie Groups? This has deep connections to physics and engineering, since Lie Groups underlie the symmetries of equations and their solutions all the way from Hamilton's equation (for ordinary mechanics) to Yang-Mills equations (for quantum fields).
I didn't find a lot of wolfram demonstrations on lie groups, so there should be some opportunity to do something a little original here. 
Some years ago, Garret Lisi created a stir with his Exceptionally Simple Theory of Everything, in which he proposed E8 as the group that embeds gravitation and all the symmetry groups of nuclear physics and electrodynamics. I know he used Mathematica to create the displays in his paper and in his TED talk. It didn't turn out to be the holy grail of physics, but it was competently proposed and a beautiful piece of work, imho.
Perhaps you could replicate Lisi's representations of E8? Perhaps you could contact him and solicit his mentoring? Perhaps this is way too much for a high-school project, but you asked for any ideas!
EDIT: As usual, Wikipedia has another nice article. This has connections to E8 and more hints that might make this project more tractable as a high-school project.
A: The site http://www.kaggle.com/ comes to mind. They have machine learning contests. For example, recently they had a contest to make predictions of outcomes of chess games - something like you learn from the outcomes of 100000 games (training dataset) and then make predictions for 10000 more games. In the paper http://arxiv.org/abs/1012.4571 the winner explains his strategy. His programs (in Mathematica, if I remember) may be available to view. You could start from there, and perhaps make improvements, or come up with a related project. Data sets are available on the kaggle website.
