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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

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    $\begingroup$ Inverse Galois problem is a major one in Galois theory. Don't expect to solve it during a PhD but I imagine some nontrivial new special cases could be. Group theory is way too broad to give a reasonable answer. One area which is still in development is geometric group theory. $\endgroup$
    – Wojowu
    Feb 6, 2021 at 15:47
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    $\begingroup$ These are reasonable questions to want to ask, but don't actually have good answers. (One of the problems is that posting such a problem as a response to this question could make it useless as a PhD project.) You should try to talk to somebody at your university who works in this area to get an idea of what doing a PhD might involve. $\endgroup$
    – Ben Barber
    Feb 6, 2021 at 15:48
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    $\begingroup$ This question is far too broad/unfocussed. $\endgroup$
    – YCor
    Feb 6, 2021 at 19:31
  • $\begingroup$ As Wojowu has said inverse Galois problem is a hard problem. As an undergrad student, I also liked Galois theory and later found out that It is interconnected with many other theories and problems in different areas of mathematics. For example, recently, of all places, I think I saw, in some other post here, that Galois theory also shows up in the theory of differential equations through something called Riemann-Hilbert correspondence. $\endgroup$
    – S.D.
    Feb 7, 2021 at 4:08

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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.

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    $\begingroup$ Similar comments apply to group theory -- maybe even more so. Most fields of mathematics use group theory in multiple ways, so if you like groups, then that is fantastic! In addition, there are multiple fields of mathematics which study groups as such -- they are distinguished by which types of groups they study, which methods they use, what sorts of questions they ask about groups, what other fields of mathematics they most closely relate to, etc. Will Sawin's advice to speak to professors at your university is applicable here too. $\endgroup$
    – Tim Campion
    Feb 6, 2021 at 17:12
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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.

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    $\begingroup$ Wow -- I was not aware of this. Assembling a list of open problems in "Group Theory" sounds like a sisyphean task. On the other hand, how are you supposed to use the book? I can't discern any organization to it -- finite groups, discrete infinite groups, topological groups, algebraic groups, Lie groups -- they're all jumbled together! And there's no discussion at the beginning of what the scope is supposed to be. $\endgroup$
    – Tim Campion
    Feb 6, 2021 at 17:21
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    $\begingroup$ @TimCampion There is one discernible organization to the book: it is a book of open problems in group theory. The scope is also clear: it is a book of open problems in group theory. I suspect this minimalist style is highly intentional, and that the authors (or, originally, the maths department) behind the book wish to leave any further organisation to derived works, not the book itself. $\endgroup$ Feb 6, 2021 at 17:50
  • $\begingroup$ @Carl-FredrikNybergBrodda Sure. It's just that if you hand me a book and tell me "Here, this is a book of open problems in group theory", I find that scarcely more informative than "Here, this is a book of open problems in mathematics." And I'd hate for anybody to pick up such a book and think there was any chance it was comprehensive or not seriously "skewed" in some field or other. I suppose they're making a statement -- claiming that it's more fruitful and useful to think about all groups from a unified perspective than I'm normally inclined to think. $\endgroup$
    – Tim Campion
    Feb 6, 2021 at 17:55
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    $\begingroup$ @TimCampion Yes, I think the Kourovka notebook is not a very good way for e.g. PhD students to learn about major trends, developments, or ideas in group theory (I agree there should be a disclaimer!). I think the value of it stems precisely from the fact that it is "timeless" (or close to being) in its approach, and treats all problems and areas as equal, letting the researchers of whatever time is reading it apply the relevant context. $\endgroup$ Feb 6, 2021 at 18:23
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    $\begingroup$ In my opinion, the fact that more than 3/4 of the problems from the first issue have been solved represents a good index of the interest that this book has in the group theory community. $\endgroup$ Feb 6, 2021 at 18:31
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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.

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