Hi,

see below how others do it https://www.liverpool.ac.uk/mathematical-sciences/research/pure-mathematics/about/

what problems in the field of pure mathematics have already been solved and what remain open.

Based on this analysis to see your area for scientific proposal.

Think about what you can do personally.

Good luck.

Mainly through a clear statement of the concepts behind your variables

Perhaps it works differently in Nigeria, but in the UK often a candidate will apply for a pre-existing project that a professor has advertised. It can be difficult for a candidate to put together a fully fledged proposal, as this requires the very research skills that they would learn during their PhD.

But if you want to put together your own proposal, the first thing to do is to identify an area of research that interests you. Importantly, there should also be researchers (with funding!) active on this area at the universities that you are interested in; if you put together a brilliant proposal, but pitch it to a researcher who has no connection with it, then it's not so hopeful.

Once you've identified the area you want to work in, read around the literature. Try to find some survey papers that give a broad overview of the field. Pay particular attention to open problems given in the survey and in the conclusions of papers.

One of my colleagues, Prof. Lazebnik, likes to quote the song of Leonard Cohen 'There is a crack in everything. That's how the light gets in' (see his paper 'a new series of dense graphs of high girth'). You need to find a 'crack' in your research area, an area that no-one has explored before or a method that no-one has tried. What happens if you try to vary the conditions slightly, imposing weaker or stronger restraints? Are there analogies with another mathematical problem that you could pursue? Or, taking objects with properties that have already been studied, can you say anything interesting about the extremal cases? Can you push an existing result just a little bit further (if a researcher has answered a question for k = 2, what can you say about k = 3)?

Best wishes,

James