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I am applying to a PhD position in Pure math in UK.

Application form asks for research proposal but I don't have any specific problem now which I would like to work on although I have a good background on the specialization I want to pursue.

I emailed my prospective supervisor the following:

I was completing the PhD application form and now there is only Funding and research proposal left. Should I fill the research proposal at this stage? I mean, I want to work under your supervision in X but certainly I am not certain what exact problem I will work on.

The background I have in X and other branches of Mathematics (including Master’s Project) has been mentioned in my CV and Personal Statement.

I have been googling about it but I don’t think I can decide what exactly I should do about it. So, can you please guide regarding what exactly should I write in it or should I leave it blank? I shall be really thankful.

His reply was following :

I am not sure about exact the requirement here, but I don't think you are supposed to know exactly what to work on at this stage. I think just an indication of your research interests should suffice.

The issue I am facing is that I don't know what problem I should work on. I could write the title as "Topics in X" and I don't know what I can write in the abstract and other parts except the topics in X I have studied and my interest in particular branches of X. Pure mathematics is really specialized.

Can you please guide how should I approach the problem I am facing?

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    I'm in Engineering not Maths, but from my experience (both when I did my PhD and now as a PhD supervisor - both in the UK), the potential supervisor would provide guidance on what to write in the research proposal such that it aligns with their research interests. But this may vary by department/university. Having a good research proposal is important if you want to be competitive for scholarships. It's unfortunate that the supervisor you contacted was not more helpful. Perhaps try to email others.
    – atom44
    Commented Jan 19, 2022 at 10:19
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    Universities in UK do not usually ask for a full research proposal, but rather a short summary of what topic/questions you are interested in (although it might be different in the university you are applying to).
    – S R Maiti
    Commented Jan 19, 2022 at 12:50
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    With the caveat that (within UK system) I have only observed things at my own university, but know them well (in mathematics) - the requirement to have a proposal is often part of the central university admissions process rather than something desired by, or important to, the selection commitee within the department. Certainly, it is recognised that applicants in your position usually won't have a focused research proposal, and so I would follow the (vague) indication of your potential supervisor
    – Yemon Choi
    Commented Jan 20, 2022 at 0:56

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This is mainly a synthesis of what's written in the comments, with some addition of my own.

It's likely that the application form is not specific to the maths department, and is used for PhD applications across the whole university. This means that the research proposal section is there for subjects in which it's more common for the student to have identified a very specific question they want to research before they apply.

This doesn't mean you should leave it blank. As your potential supervisor suggested, you should write about your research interests and give some evidence that backs up that interest, as well as a reason why you want to study that topic at that university.

For example, you may write something like

I am interested in pursuing research in the topic of Advanced Y Problems. In my undergraduate degree, I greatly enjoyed the course on "Introduction to X", in which I achieved a grade of 99%*. This led me to doing a summer project studying Applications of X to Y. During that project I found that X can often be applied to Y, which built on the work done by NBH et al. (1995). I am keen to continue studying X and Y, and especially Advanced Y Problems, due to my strong background in Y from the courses "Topics in Y" and "Topics in Y Part 2" which I took in the final year of my degree. I have chosen to apply to the mathematics department at the University of A because of the active research group in X, Y and Z and especially because I am interested in studying if there is a link between the applications of X to Y and Y to Z, as recently proposed by MTW et al. (2019).

Obviously you would give more details on the particular topic you're interested in, courses you've taken related to that, any projects you have done, papers you have read and so on. I would aim to write about a page in total.

Good luck!

*This is not a suggestion that you need such a high grade to succeed in PhD applications! Just an example :)

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