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I'm thinking of applying for a summer scholarship (in dec) from our maths department where we undergo study for 8 weeks with a project in maths. We have to have our proposed topics in by 7/Sept but I'm really stuck for ideas.

I'm a freshman at my university, and have taken/taking:

the usual calculus (not to Vector calculus though, that's next year),

a course on proof theory (principles of math),

discrete math (set theory/logic/simple combinatorics)

into. to applied and computational math (matlab/ODEs/various numerical methods).

I'm also auditing a senior course on Algebra (groups/rings/fields/extensions/all the juice bits),

and after the mid-semester break i'll be auditing a sophomore logic course and studying things like formal logic systems/decidability/completeness. One of my friends is a senior logic student and is an excellent mentor.

I want to do a small project involving field theory, but can't think of any topic that I could do. I think they expect it to be original, and my inability to choose a topic probably means that I don't know enough to be doing it in the first place. But I really want to get stuck in, and have been studying the topics involved.

A couple of people in my calculus class are doing it too, although I'm not sure what in (something with physics). I'm going to approach my Algebra lecturer but his next available office hours are really close to the deadline.

I think being tossed into the deep end, will at least, let me into the maths department and meet people who I wish to spend a lot more time with in the future years. But also, if I do badly it might stop me from being able to do it next summer, when I will be more mature mathematically.