Okay, so I know MO has had a recent proliferation of this kind of question, and I know MO is not really for this type of question (though I suspect perhaps this is a phenomenon that is likely to repeat toward the end of every academic year...)- nonetheless I find myself cap in hand and hoping for some guidance.
I wasted my undergraduate degree: following a fairly successful first year and an interest in pretty pictures, I found myself digging around in the region of complex dynamics and fell for it hard. As first loves go it was a great one- I swooned over Montel's theorem and cooed over the simple presentations of iterative dynamics gleaned from the uniformization theorem- but like all first loves; the detail of the thing did not surpass the idea, and pretty soon it had to end. I was disillusioned and reluctant to look for more fish in the sea- my work ethic dropped to zero.
I fell in love again: but too late- algebraic topology/ differential geometry hit me in my fourth year like a simplicial arrow from cupid's own bow but by this time, my grades blew and all the people I knew in the department were DS theorists. I got a 2:1 (for all you non-UK MOers- it's a degree class that's basically a rubber stamp with the word 'mediocre' on it).
I tried teaching school kids: not enough cohomology.
I've got myself a year, a jolly good library and a lot of determination: My aim being to produce something so intriguing/charming/advanced that someone will give me funding to do pure maths.
So what, if anything, should I try to produce?
Specifically: Would I have to solve some grand unsolved problem? Would I get by with just a small one? If so, where would I find it? Perhaps even a complete set of excercises from an advanced book? A digest paper on a difficult topic? [If it helps my research interests are differential geometry, differential topology and gauge theory- but I'm flexible]
I am aware: That the above situation is my fault- and I would be grateful if you were restrained in your remonstrations. That the question, as stated, is highly subjective- but the opinions of research mathematicians is precisely what I am trying to gauge. That the answer may simply be: 'try some less prestigious universities'- in which case, fair enough- but I don't want to rule anything out just yet.
Thanks in advance for any help you can spare.