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.
Background
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.
Question
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.
Best Answer
I sympathize with your case. A 2.1 is really not bad. You shouldn't denigrate yourself and view your peripatetic interests as requiring redemption.
Taking on a big unsolved problem without guidance or the background of a PhD student seems doomed to fail. Locking yourself in a library with all the world's books is unlikely to produce anything of merit. I have never heard of a case of a student producing something of "intriguing/charming/advanced" and using that to gain graduate admission. The romantic, amateur heroic view of math is largely bunk as pointed out by Terry Tao.
There is still hope. I know that undergraduate research is less common in the UK, but I would expect that if you email lots of professors in areas of interest to you and basically offer yourself as cheap or free labor (undergrad student level), there is a good chance that you'll be taken on as an unofficial research student by someone. I know many cases of people in math and science using this sort of informal contact to start research projects that eventually develop into PhD positions. Making yourself known to a tenured professor who can write you a strong recommendation is probably enough to get you a PhD position somewhere (in the US, UK or Europe). It is unlikely that claiming to solve a big problem or do research on your own is going to be trusted by graduate committees. You need recommendations from people trusted in the academic community.
There are MO users who have taken a decade or more off from education and successfully started PhD positions at Princeton and other top research institues. Good luck.