How to narrow down a research topic in mathematics?
Short Answer Don't be too picky at first, learn about others' work; try to come up with some problem &lt;-> solution pairs; and then you shall find your own path; when writing them down.

Longer Answer

Fun Fact: Publishable results are based on published results and some more. You need to do 'the some more' part, and know the 'published results'. So: 

1. High Rank Publications and their citations: Find the best publications in the area of your work, and see how they solve the problems and how the present their results and who they cited; and read and learn them as well.

2. Narrow Down More and More When Writing Down: Ok, now after reading some papers/journals/books, you want to write something. Here we do the following: 

2.1 Problem and Solution (Potential): You might ask what problem I want to work on? Well the one you know how to solve! I might go and write a code generator and say: "well, it is fast and 'efficient'". How do I know? Well I don't know! I need to check it with others. So, we go to the next phase: 

2.2 Evaluation and Background Checks: You start evaluation of your work by others' approach (and write about it). Must likely your approach is not good enough, so you go back to he first step and update this one. 

2.3 Introduction and Evaluation Now you know what you are talking about, you write the introduction and conclusion about it. 

Then Boom! you have a paper, that could be published. Find a conference/journal and hope for the best!