Writing a research statement in mathematics without having any research experience
Your research statement should set out the background to the problems that interest you, what you think you could contribute to their analysis/solution, and an overview of the approach(es) that you anticipate that you might have to take. People want to be reassured that you understand the nature of research (ie. how it differs to an undergraduate programme of work), that you have a substantial and interesting problem in mind, and that you are potentially equipped to make an original and nontrivial contribution to knowledge.