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.