From discussions about the nature of pure mathematics research I reached to a conclusion that it requires both- theory building; that is to create rigourous concepts relevant to the problem one working on and then building upon it, as well as usage of clever, ingenious tricks to obtain certain results. There's no real dichotomy in here.

If this is the case then I'd like to know how can I do certain things or develop a few habits that will help me or atleast equip me to tackle the challenges professional pure mathematics research poses before an individual?


Arguably, doing a PHD is supposed to be exactly the preparation you speak of. It is the first time in your education when you are most certainly expected to transform from the mere consumption of Mathematics to also producing new results; the general idea, however, is not that you simply start doing so, but rather that you have an advisor who guides this process. I mention this because you'll do well to remember that your advisor has this responsibility towards you, as you have the responsibility to invest time and effort to make the transformation, which is not an easy one for most people.

That said, here is my list of tips:

  1. You may find yourself under a lot of stress to produce results. Unless (maybe even if!) you are an absolute one in a billion genius, the results may take quite a while to come, months, maybe years. Prepare yourself mentally for this fact. Although, I am afraid, I was told the same thing in advance, took it serious, and I still was absolutely unprepared for the reality of it.
  2. "Having a new idea" is what you rarely learn in undergrad courses, where most people puzzle together theorem statements from the lecture to prove exercise results. This is not a bad thing, you will need this skill to make your ideas work, and you will also need this skill to get into new fields where new ideas are needed. However, having an idea is different and how exactly you come to have ideas, noone can tell you. I have known many different mathematicians and their minds all work in different ways. However, there is one thing they all have in common: They have their best ideas when they are relaxed and at ease. Find ways and time to be relaxed and ponder about Mathematics without feeling stressed out. How exactly you do that will be up to you.
  3. Work with other people whenever you can. Collaboration is extremely powerful both to keep you engaged with a problem and to overcome obstacles.
  4. Be enthusiastic or have the ability to get enthusiastic about your research topic. Without that, you are unlikely to muster the mental endurance required to actually do research in it.
  5. At the beginning of your PHD, you must listen very carefully to your advisor. Towards the end of it, you must make sure that your advisor is listening very carefully to you.
  6. Make concrete examples. I explained this at length in a Math.SE answer. This should work even in very pure fields, even though "concrete" may mean something else for you than for others. The important part will be that it is concrete for you, i.e. that it enriches your intuition.
  7. Find a good balance between intuition and rigor. Relying too heavily on either will get you stuck. I believe that the more experience you have, the more you can rely on your intuition; but for someone starting out in Mathematics, I believe it is healthy to go with a 50/50 approach here.

While it is not my tip, I would like to include this excellent suggestion from the comments:

Keep a notebook of ideas that might result in fruitful explorations. Review it periodically. These are the ideas that you don't have time to develop now, but might lead to something when current projects come to fruition.

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    Nicely said. I like 5. and 7. especially. Hell, all of it, especially. – Buffy Jan 26 at 11:57
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    I would add: keep a notebook of ideas that might result in fruitful explorations. Review it periodically. These are the ideas that you don't have time to develop now, but might lead to something when current projects come to fruition. – Buffy Jan 26 at 12:15
  • Thank you for the kind words! I have included your excellent suggestion in the answer to make it a little more visible. – Jesko Hüttenhain Jan 27 at 18:31
  • Rigour is something necessary and obvious for someone to develop to even do stuff on usual basis, how do then one should approach developing intuition? – S_Mitter Jan 28 at 4:34
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    @S_Mitter: The three things I know that have been proven to work are practice, practice and practice. A lot of intuition is just having done something similar many times before. But while you do so, always try to keep in mind the why, i.e. don't blindly smash formulas together until you arrive at a proof, but always try to think about it on a more general level. E.g. for a longer proof, try to summarize it in a few informal but useful sentences. Intuition often is just thinking of those few informal steps, while at the same time being practiced enough to know you can execute them. – mlk Jan 28 at 9:09

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