research statement/proposal for a (pure-)math-PhD?

Question

When applying for PhD programs outside your home university, the university you apply to usually requires a research statement from you.

However, this isn't the case when you reach out for a Bachelor's or Master's thesis topic to a professor you know from your home university (in mathematics). You would just approach a supervisor and express your interest in certain topics. They would provide you with various directions to choose from, and the exact problem you would work on may crystallize while working on your thesis. The same principle applies if you pursue your PhD at your home uni.

Nevertheless, when applying somewhere else outside your home uni, they will ask you for a statement.

How should this statement look like?

Typically, most students completing their master's thesis have no exact idea of what they will do in their PhD. They may only have a vague notion of the topics they like or an unspecific direction in which they want to focus.

I mean, how can they ever come up with a detailed research proposal?

Especially in pure math.

Since research in (pure) math is like being a blind man, in a dark room, looking for a black cat, which isn't there.

Answer 1

In my experience, a research statement has the intention to be a formal declaration of interest. The way in that this may be made is, at the beginning talk about you, your academic background and specific knowledge focus areas, and, in accordance with the field of the receiver, use the rest of the statement to prove knowledge about the work that is developed by and how your background and expertise can contribute to solving problems in the research. In that way can you demonstrate knowledge and interest more than a random search for opportunities.