How do you go about doing mathematics on a day to day basis?

The method is "keep trying". Try reading the paper again, try reading a different paper, try stopping reading papers and just work it out yourself. Try talking to other people, try struggling on your own, then try talking to people again. Try a different problem, then try going back to the same problem. Try the same method again and again, then try different methods, then go back to the first method and try it one more time. Try just writing out some calculations even if you don't see them going anywhere, try a stupid example, try drawing some pictures. Finally try taking a break, and then also try not taking a break. Try everything once, then try it again!

But enough of that, here's some more practical advice from my own experience (I'm a postdoc by the way).

1) For a hard problem, work intensely on it for a few weeks or months, until you feel like you've completely hit a wall. During this period you want to be producing as many calculations, lemmas, pictures, examples, etc as possible. They don't all have to be super-relevant -- your goal is maximum output of material. Then when you finally feel like you're going insane, take a deep breath and move on to something else. After a break of a few weeks or months, which may be spent either on work of some other kind or even on vacation, come back to the problem and see if the fresh perspective helps. It may take multiple cycles like this for a really hard problem.

2) Look for a variation of your problem which is easier or which your methods can better handle. Even if the variation is not as strong of a result as what you are ultimately aiming for, it may still "count" and end up being a nicer paper than you realize at first. It's good for your morale to register progress in this way, and if you can get a publication it's good for your CV too. Here is where advisors and mentors can help a lot -- they can help you identify which variation is meaningful, doable, and "interesting".

3) Remember that almost nothing ever works. One of the worst feelings to me is when I find a mistake in something I thought I had figured out, and all of a sudden weeks or months of progress begin to unravel. This can be extremely depressing. Try not to take it too hard, and over time you can rebuild and recover. Try also to appreciate the progress you are making, even if it is much less than you wish it was.

Good luck!

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