As my question implies, I am a young, developing mathematician, with concerns about my future math career. In particular, I'd like to know how to select a future field of study. Through my courses and readings, I've started to entertain the possibility of studying Lie theory. I think this field has some nice, deep structure and is particularly well-suited to my minor interest in physics. Of course, other fields are also appealing, so I am having a hard time narrowing my choices.
My question is, then, how does one select a field of study? I'd love to hear some personal experiences that relate to this question.
Some of the considerations for choosing a field of study, which I have been asking myself, are
- What appeals to you? What do you enjoy doing?
- What do you want to know? What questions do you want to answer for yourself?
- What does the world want to know? What fields have important, pending questions that should be answered?
- To what fields is your chosen field connected? Do these connections play a large role in your chosen field? Are you interested in these possible connections?
- Which fields are are on the rise? Which fields are currently exciting and moving in a good direction?
- What are you good at? What areas seem to be intuitive to you?
If someone could talk about some of these considerations in regard to their own experience, I would be particularly grateful. Furthermore, if there are any considerations that I am missing, please feel free to add to the list.
Best Answer
Don't worry about what fields are "hot" or not (point 5). For one thing, that can change very quickly when you least expect it. New results or sudden applications may catapult a particular area to the fore, and answers to questions can doom a field to obscurity. It's going to take you several years from now until you are done, and there is no guarantee that what is "hot" or "on the rise" right now will still be hot or on the rise when you are done.
I would also advice to give preponderance to 1 and 2. If you become a mathematician, you will spend your time thinking about these things, and they better be things that you want to spend time thinking about. If doing your research becomes a chore, you are doomed, no matter how good you are at it, or how hot the field is. You'll be looking for excuses not to do it.
Point 4 is not really something that can go into deciding what field to study, because you sort of need to know a bit more about the field to really be able to answer it.
Mainly, in my opinion, you'll want to do something that you personally find exciting, interesting, and fun, for whatever reason. Sure, it's nice for the people who are working in hot areas and can easily get grants, since it is always better to be rich and healthy than sick and poor, but remember: if things go well, you'll be doing this for 30-50 years. You want to enjoy it.