The best kind of proposal is for a program
you want to make, something that will challenge you to learn something new but be mostly based in what you already know.
For example, a final year University project proposal might actually be
- a small, garbage-collected language. Or
- a fancy GUI that creates and manipulates varieties of Turing Machines and visually executes them for classroom use. (That's what I did.)
- A friend of mine wrote something that played with the properties of prime numbers.
- Even a simple, networked video game that requires you to apply disciplines in memory management, databases, networking, and UI design. (I nearly finished a Pente game that students can play on the school's servers [unfinished because it was a project for teaching, and the class ended before it could do more than handle players at one terminal].)
- The
PLIS Group has a lot of mathematically-oriented ideas here:
http://akira.ruc.dk/~mir/plis/studentprojects/
If this is only your first or second year, try something
small. You don't know enough yet to make a Quake engine. Remember, the purpose of a project proposal is two-fold: (1) demonstrate what you know, and (2) demonstrate that you can learn and apply something new.
So you can do things like
- make a 3D tick-tack-toe game, or a breakout clone, etc.
- do the Knight's Tour with some look-ahead intelligence.
- write a little graphical demonstration of Maxwell's Demon.
- reinvent some wheel. (Write a "make" program, or a file and directory manager, or a program preprocessor, a spell-checker, word-counter, etc)
In short: Make a list of some things you would like to do. Examine what you would need to know to do each one. Discard those that are too easy or that would require you to learn a whole new discipline. Pick one that you can use what you know to learn something new (something small).
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Sometimes (particularly if you are new at this) it is hard to know where your limits are. That is where consultation with your professor comes in. Narrow your ideas down to two or three and ask him to help you decide on which one to implement and what requirements (or goals) you would like to meet with it.
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Good luck!