Writing a research proposal in mathematics for graduate school My question is about research proposal, I want to apply for a PhD program which requires writing a research proposal.

What exactly should I write for the methodology section of a research proposal in the area of mathematics? I have several ideas about my project, but I have not proved anything yet, so how can I know its methodology?
My recommendation is to try to ask some current students if you can see their proposals.  Obviously, what they did can't be all that terrible.  

More generally, I would agree that methodology is not a extremely well defined concept in mathematics, but you can write something intelligent about it anyways.  How are you going to approach the problem?  Are there special cases that might be illuminating?  Stepping stone lemmata you might get to before the whole proof?  What areas of mathematics do you hope to use?  Unfortunately, I can't speak for the system in Italy, but certainly in the US with such research proposals, the main point is to see if the candidate can write about and analyze mathematics in a cogent way, not the details of the problem.