Differential Geometry and Differential /Algebraic Topology at first is a wide field. The main problem will be that you have to read the books and work the examples. Simply by reading (personal experience!) you won't have any chance to reflect the problem unless you are some sort of a genius. But as was pointed out before this is a rather romantic view of mathematicians althought there are certainly some exceptions from the general case. Secondly, I don't think that a big unsolved question can be solved offhand by someone with no PhD background knowledge. Or even if someone has that knowledge: mathematics is not made by persons sitting alone in a poorly lightened chamber with tons of papers and thousands of pens. No, it is made by people going to conferences and symposia or what else. The great theorems in our times mostly (I know there are exceptions like Perelman) deduced and proved in a team. Consider for instance the Atiyah-Singer Index Theorem. As the name already suggests, it was made up by two of the brightest mathematicians of our times. The situation in physics is quite the same.
BUT: If you really want to conduct research and not only read mathematical texts because it seems like an alternative to thrilling novels to you AND you already have a college degree, enroll in a graduate program. If you're motivated enough and succeed in showing the profesors that, I think you will have no problem with nor being accepted. As was already pointed out, there are many examples of people who enrolles 10 years after completeing their bachelor's degree in a graduate program. And they were successfull overall.
But I don't like the idea of giving someone something to conduct research about. Research is all about finding your personal interest and your personal style. And again, it's completely unrealistic, in my opionion, that big problems are solved overnight by people who are not directly involved in the mainstream mathematics (at least officially). Although, admittedly, there are exceptions (Perelmann, e.g., but even he completed his mathematical studies before that).