How do you approach doing research on an unsolved mathematics problem?

Question

I have difficulty gathering all the information I need on open questions. How do I obtain all of the historical and technical details of a problem (e.g. a millennium problem)?

Answer 1

Take a three-pronged approach.

1. Find important articles in the literature of that question (for example the article that originally posed it or major milestone papers solving related questions) and then read every article that cites them.

2. Find and read survey articles that introduce you to the field and the literature.

3. Read newly published research papers and preprints.

These three approaches will reinforce each other. You'll find survey articles and lit reviews, as well as recent papers, when you look up all the articles that cite the milestone papers. Meanwhile, the survey articles and lit reviews in new papers will direct you to important papers in the field.

Don't take papers for granted. They're written by people, who make mistakes. Work through the arguments and prove the theorems yourself. Whenever you get stuck, take the argument you don't understand and find textbooks that walk through it and early papers that introduced it.

Don't be afraid to ask mentors, professors, and colleagues for help if you have a question and they're an expert in the area. If you're respectful and smart about it, this is a good way to start to make connections and learn who knows what in the field.

After months of this (or years, if you're jumping on a big famous problem like one of the Millennium problems) you'll see patterns, the same themes and ideas and papers emerging over and over again. You'll also get a sense of what's "hard" and what's "easy," which problems will require a lot of work and new ideas to solve and which problems might succumb to known techniques with the right perspective and calculations.

As this happens you'll start to have ideas about how to extend or refine various ideas and arguments that you've seen. Be sure to talk about these ideas with other people you trust in the field -- maybe they're actually dead ends, or maybe they're new and you'll find some collaborators.

At this point, congratulations: You're ready to start contributing!