This section is devoted to advice and wisdom I've collected pertaining to doing [mathematical] research. Feel free to disagree on the importance of what I say as it is only my opinion!
- Have an understanding of what problems are currently being worked on in your field of interest. You need a place to start after all! Save the citations you get from the papers you look over so you have them on hand when it comes time to write a paper [introduction].
- Get familiar with Python, Matlab, or Mathematica if you aren't already. All of these languages are useful for testing conjectures and running algorithms you hope to turn into theorems. Comment any code you write so you remember what it does later.
- When communicating with colleagues over email, use descriptive subject lines. Not only is this a courtesy in this kind of professional environment, but it can help you find the discussions you have to refer to them later
- Double-check your work meticulously. Gaps in reasoning can exist, but more frequently I find errors like dropping a constant, swapping a + for a -, etc.
- Keep track of what papers you have read on a subject; in-depth projects may require backtracking and have you searching through the literature in several iterations [this piece of advice is courtesy of Nicholas J. Higham].
- Try to present your work at least once. The reasons for this are many. The obvious one is to hone your presentation skills, but it also helps test your understanding of the material, especially when audience members ask you questions. Moreover, an insightful audience member might make comments that take your project in directions you never thought existed. Along these lines this is a great means of facilitating collaboration and networking.
- Be sure to rehearse your presentation. Bonus points if you can do it in the room in which you will actually present.
- Always look for ways to generalize what work you've done. This usually involves either removing a constraint or working the problem in a higher dimension. Along these lines, the one and two-dimensional cases of problems are usually the most straightforward and might lead to inductive arguments for higher dimensions.
- Be ready to ask around for textbook suggestions in the event you need more background on a particular topic. I think this one is especially prevalent for researchers still early in their careers.
- If you are the recipient of a referee report or some other list of points of feedback that refer to specific places in your paper, address those points in the reverse order in which they are written. This is because the comments are generally listed in chronological order with respect to the paper, so addressing the comments in that order may cause line or page numbers for the later comments to shift.
- If you are reading papers and have a comment or question about it, the authors' email addresses are listed for a reason. Don't be afraid to reach out.
- Use databases such as Arxiv and MathSciNet to scope out what research has already been done, especially if you have a particular author in mind.
- If a question seems easy, there are two likely reasons why: you are forgetting an important aspect of the problem, or it has already been solved. This isn't ALWAYS the case, but it's kind of how research progresses; each result builds on older ones.
- Have backups of your slides in case something goes wrong during your presentations: USB drive, post your slides online, email them to yourself, etc.
- Put your contact information on your last slide so people can easily email you if they develop questions near the end of the talk (as opposed to just having them at the beginning)
- A fresh look at a problem after a good night's rest often leads to new realizations!
Now, here are some of my favorite quotes about math/professional development/problem solving.
- “If you want to get very good at something, find someone better than you and learn everything you can from them.” (Jonathan Little)
- “Play is the ultimate form of research." (Albert Einstein)
- “When people talk, listen completely. Don't be thinking what you're going to say. Most people never listen. Nor do they observe. You should be able to go into a room and when you come out, know everything that you saw there and not only that. If that room gave you any feeling you should know exactly what it was that gave you that feeling. Try that for practice.” (Ernest Hemingway)
- “Research is the ultimate open-book test.” (This one is mine)