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?

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    What does the term "methodology" even refer to in mathematics in general? This sounds like it might be a standard form meant to be used for all subjects. Mar 16 '14 at 14:34
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    You can try to explain with which techniques you believe you can solve the problem you pose. It does not matter that you have proven that these work, but you should try to convince the reader that it is a plausible approach. Mar 16 '14 at 15:26
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    What sort of Ph.D. program is this? (E.g., which country is it located in? Is it in a mathematics department per se, or another department?) I've never heard of someone having to include a methodology section in a math grad school application, so I assume it's a different sort of program than what I'm familiar with. Some more details about the program might make it easier for someone to answer. Mar 16 '14 at 16:21
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    Is a methodology section explicitly required? Even if it is standard in other fields, it's possible that most mathematics proposals would not include one. Mar 16 '14 at 17:10
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    @AnonymousMathematician From my experience this past application season, required research proposals were standard in graduate applications in Europe (for math). I never saw one that required a methodology section, though.
    – user10060
    Mar 16 '14 at 19:16

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.

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