This is probably going to be a long post, but its something I've always wanted to know and I believe many could profit from it. How do you determine a field of research?

Of course, this question can be asked for any subject, be it Math, History, Physics, etc. But in Math, this is an even bigger problem, in my opinion: the field is just so massive! It has been around for millennia, and thus it is pretty much impossible to get to know every field of research before you make a decision.

The natural suggestion is to then narrow your scope - look around the major areas: Algebra, Number Theory, Topology, Geometry, Analysis and Foundations (this is, of course, one of the multiple ways to split math into different subfields, and is definitely not even close to perfection) and find the one you like best.

But even this still leaves way too much room. For instance, I really fell in love with Algebra after my first contact with it, studying Rings, Fields, Groups, and all that good stuff. But how am I supposed to know which subareas are there and what questions they try to answer?

I know about a few, like Field Theory, Ring Theory, Group Theory, Representation Theory, and Galois theory. What other areas are there?

And even in those areas which I've heard of I can't state more than a handful of problems - and that's not to even mention subareas which are on the boundary of Algebra and have a huge overlap with other fields, such as Algebraic Geometry and Algebraic Topology (again, I know the division is artificial, but it helps me state my point).

In my university, the Algebra department focuses on a narrow range of topics - what if there is something outside that range that would capture my heart, but I haven't even heard about it?

And finally, what if I go into an area thinking it will be amazing and then lose interest? Can you change your course if you make a wrong call?

I'm sorry for the long post, but it is really difficult to take all those things into consideration - to grasp the broadness of math as a future mathematician. If anyone has been through this, please, share how you dealt with these questions, I would love to read it! And, if you could give me some references as to where I can find out about all these areas of research, what they study and that sort of thing (especially in Algebra), I'd greatly appreciate!

Thank you so much in advance!

OBS: I have read a few other SE posts, such as How to select an field of study? and What made you choose your research field?, but they haven't really answered my question, so I decided to write my own post.

  • $\begingroup$ Is a field of research "determined" other than by your own interests and aptitudes? Graduate programs help to refine your vision, but the reality that the body of mathematical knowledge is vast should not necessarily hinder your research pursuits. $\endgroup$
    – hardmath
    Commented Aug 31, 2020 at 23:05
  • $\begingroup$ I think a lot of finding your niche is trial/error and guidance from veterans in the field. As you work to learn different areas of algebra, you'll become aware of more subfields. It may be helpful to you to casually browse arxiv.org. Don't expect to understand everything, but it will give you an idea about different areas of active research. $\endgroup$ Commented Aug 31, 2020 at 23:27


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