What are the main open problems in Group Theory and Galois Theory? I’m really interested in doing a PhD and the subjects I enjoyed the most were Group Theory and Galois Theory.
What are some open problems in these areas that would be suitable for a PhD? Is Galois Theory so niche that you couldn’t do a PhD in that area?
Thank you
 A: I also agree with the comment of Ben Barber. In any case, let me mention that a good collection of problems in group theory is the Kourovka Notebook.
A: If a problem is an (important) open, then one can argue that it is not suitable for a PhD—because on the one hand, it is, generally speaking, not a wise decision to attempt an open problem for a PhD; on the other hand, if you could solve it, chances are that experts could solve it already too.
Better consider working with an advisor in group/Galois theory or a closely-related field; in the process, you would make a significant contribution, or you might chance upon a novel/groundbreaking idea that could resolve an (important) open problem.
A: I agree with Ben Barber that you should not try to choose a PhD problem by picking from a list on MO, or anywhere else on the public Internet. However, there is one useful non-obvious piece of information we can give you:
There are a lot of mathematicians doing research that heavily involves Galois theory, however, very few of them would call their research area Galois theory.
In fact, almost every subfield of algebraic number theory heavily involves Galois theory in one way or another. Professors who describe their research area as "algebraic number theory", "Galois representations", "Iwasawa theory", or "the Langlands program", to name a few possibilities, could most likely give you a problem heavily involving Galois theory if you were their PhD student.
You could therefore apply to a PhD position at schools strong in number theory. You should speak to professors at your own university, in particular ones who might be writing you recommendation letters, to decide which schools in particular you should apply to.
