Im interested in differential geometry (as well as algebraic geometry) and all forms of topology mostly, but i dont know whether I will be capable of research because i am unaware of what has not been researched already and if there is anything to find.The answer to your question is very individual...
There are many beautiful subjects in math, such as (for my opinion): Game theory, Additive number theory, Knot theory, algebraic geometry, algebraic structures, combinatorics, complex analysis, etc...
Maybe if you'll be able to tell us what are you interested at, we'll be able to give you names of fields related to your answer...
especially since you seem to be 17 years old, could you be more specific about what your math level is right now? What courses have you taken, what degree do you have?also you didn't specify 5 years of what maths.
a) Some schools let you test out of pre-reqs; contact the ones you plan to go to.Because i am applying for university at the end of next year, i would like to write a paper to use as substantial evidence to put me through an accelerated program because of course doing at least the first 3 years of mathematics again would be horrifying