I got my Master's degree in mathematics in 2017. Since then, I haven't done a lot of maths, so I am a bit rusty. I am now considering doing a PhD in maths, but I have little-to-no idea for a topic.

I seem to understand that it is not unusual for prospect PhD students to be without a topic at first and be sort of "assigned" one by their supervisor, but I think I would nonetheless benefit from having at least some sort of idea. Ever since I picked up math, I think I've been more drawn to more abstract and theoretical math and not at all to applied math. I am fascinated by number theory problems, and I like algebra so I thought I could do something related to them. My bachelor's thesis was on algebraic topology and my master's thesis about wreath products and the Rubik's cube. However, PhD my topic doesn't necessarily have to be related to these. I can't say I have any preference at this stage, and I'm trying to find one.

I tried to get some inspiration by going to the campus library and checking out different books, but that didn't work well for me. Most books are extremely technical and you can't just flick through one to get a sense of whether you like the subject or not. (More often than not, the title won't even tell you if the book is worth picking up.) I tried googling, reading Wikipedia, etc, but I can't say that helped, either.

What I'm after is some sort of overview that presents problems in different fields, explaining why they're important/intriguing, where we're stuck with them, pointing out connections with other problems or branches of maths. Something that tries to "sell" you a topic, if you will.

I'm not sure such a thing even exists, so any suggestion you might have to help me get an idea will be much appreciated.

  • 1
    Are you currently in a doctoral program?
    – Buffy
    Sep 21, 2022 at 14:17
  • @Buffy No, not yet
    – Nicola
    Sep 22, 2022 at 10:49
  • As a clarification are you looking around to try and work out what kind of supervisor you want, i.e. you want to hear about a lot of different fields of maths so you can be certain with what field you want your supervisor to be from? Sep 22, 2022 at 15:23
  • @NAMcMahon Sort of. My options are limited because I am in no position to travel, nor do I want to. There are two universities that I have access to where I live, so those are my options, and I'm starting just now to get in touch with people there. I know many will say that if I don't want to travel, that's a problem, but that's simply a non-negotiable in my situation.
    – Nicola
    Sep 23, 2022 at 14:17

3 Answers 3


In most places it is too early for you to commit to any particular thing. That would be especially true in the US, but also other places to a lesser degree.

I suggest that, at the moment you think about what sorts of math you enjoy doing and have some interest and insight into. It need not be terribly specific. Maybe number theory in your case or some subfield of that. My own narrowing led me to Real Analysis and/or Point-Set Topology (yes, long ago and far away). I ended up with a really nice dissertation in Real Analysis (Last of His Kind).

Next find a university that has a few faculty members with similar interests as evidenced by their publications. Schools with large faculty may automatically serve in some cases, but having a weekly or so "seminar" in some related area is a good sign. Apply there and get accepted.

Now you are in a position to sit down and talk to people in the area who can describe things that you might find interesting. If there are three or four faculty there then you have a good chance of getting inspiration from at least one of them. I talked to a couple of people who claimed not to have any fresh ideas at the time, but recommended me to the "senior" member of the group who took me on.

Some places (US) this is easy to do since the program is a bit longer than others. In other places you need to do this remotely since you may need to find an advisor to get accepted to the program.

But the best source of inspiration is to be somewhat adjacent to a person who can help you.

Mathematics itself is huge. There are a huge number of possibilities. You can spend too much time searching in this vast space being intrigued by many things. But a knowledgeable person can help you focus, provided that you can narrow the field somewhat. Collaboration is a really good thing in math. Sharing ideas. Inspiring one another.

I'll also note that if your proposed topic is too specific and if you are too insistent on that and that alone, it will be harder to find an advisor/supervisor interested enough to take you on. You need some flexibility at this stage and the willingness to compromise enough to get through a degree. There is no need to do something you'd hate, though. But, math is, like I said, huge.

  • 1
    Disclaimer: any field of knowledge is huge, when you need to pinpoint a theme for your PhD. I always tell people that while a Master thesis is 6-12 months work focused on one curricula exam, a PhD is 3-4 years of work focused on one topic of one curricula exam.
    – EarlGrey
    Sep 23, 2022 at 9:47

I think a main source of inspiration can be other academic studies. Particularly in the conclusion part will be a section or a paragraph, that includes suggestions for further/ future studies. You can use these sections to find new topics or advice.


tl;dr: You just need to read through recently published PhD thesis. If this does not help, if it does not raise minimum interest and "I can do that" thoughts in you, nor motivate you, then you are done: a PhD is not for you ;)

In research it is often oversold the concept of novelty. And from the exterior, it may even exist the myth of the romantic researcher battling their demons and developing their ideas out of hard work and thin air.

In reality, we stand on the shoulder of giants(Lincoln, 1905). In fact, the capital sin in science is copying without referencing. An even worse sin is referencing without copying, reinventing the wheels and wasting resources on the way.

Therefore, if you want to draw inspiration for your PhD, just look for recent PhD thesis. It will give you a glimpe of the work that expect you. To give a broad approximation, for every written page you read in a PhD thesis, assume the author wrote 20 pages of notes and probably read (or at least skimmed through) 400 pages of papers, books, conference proceedings, preprints ... then ask yourself: is it really what I want for the next 4 years of my life?

  • 1
    >! The answer may be yes or no, don't worry :)
    – EarlGrey
    Sep 21, 2022 at 19:58
  • 1
    That doesn't answer my question. I did not ask about the kind of work that expects me, I asked about possible ways to find out what topic might interest me without having to read through entire books first.
    – Nicola
    Sep 22, 2022 at 10:52
  • tl;dr: You just need to read through PhD thesis! If this does not help, or intrigue nor motivate, you, then you are done: a PhD is not for you ;)
    – EarlGrey
    Sep 22, 2022 at 11:46

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