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I am a math PhD student and I should finish the PhD in about 1.5-2 years. I am already thinking about what to do next. At the moment I am not even sure that I will be staying in the academic world, since I found out that some industry jobs can be quite fun and challenging too, but we'll see. Anyway, I am a bit worried that by the time I graduate I should have 3-4 publications on a specific topic, i.e. the topic of the PhD thesis, but after these papers are done I am not sure on how to proceed if I wanted to pursue an academic career.

I mean, I know what some interesting questions left open in the sub-sub-topic I picked are, but, except one or two, the other ones seem way too hard for me to tackle, in all honesty. So I am wondering: what is expected from a post doc? I can't see myself in two years to be that much more knowledgeable or skilled than I am now, and changing topic is a bit scary: after all it took quite some time to do research in the topic I chose for the PhD, so if I had to start over with a different (although related) topic I would need some time to adjust, and the time needed may be highly variable depending on the topic.

Also, I was wondering if in principle it is better to try to find a post doc where you can basically continue your PhD thesis or to change topic afterwards. I know I'll have to do that eventually, but maybe for the first post doc it's more common/better to keep the same research topic of the PhD? I don't know. I can see pros and cons with both options and I am not really sure what's the best bet here. Any advice from more senior mathematicians would be really appreciated.

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    A post-doc should show that you can move on from your thesis and do something new. Two years from now you should be much much more skilled at math than you are now. Productivity of a graduate student looks like an exponential curve as your learning and experience compound with each other.
    – Jon Custer
    Commented May 22 at 23:10
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    Keep in mind that you will generally work in the same field as your PhD, just perhaps not the exact same problems. Most people in pure math don't change fields from PhD to postdoc. Commented May 23 at 0:22
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    You have to be realistic here: unless you are extremely strong, you will be lucky to get 1 or 2 job offers and choosing between the two might be based on very different factors (location, salary, teaching load, prestige...). Commented May 23 at 15:17
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    " I can't see myself in two years to be that much more knowledgeable or skilled than I am now" are you sure? In hindsight my academic learning has followed something of a geometric growth. 2 years before I graduated from my PhD I knew nothing comparing to now. Commented Jun 2 at 15:04

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Math postdocs should generally spread their wings and explore new topics within their subfield. This is where mentorship is important. If you have a formal postdoc mentor, that person should help you select/develop good problems to work on, that are relevant to your prior experience but different enough that your range of expertise improves. In addition to that, you can talk informally to senior people in your area about research problems. You can even ask people directly whether they think problem X is a good idea for you to work on. (How helpful of a response you'll get may depend somewhat on your relationship with the senior person in question. But I did find this approach useful overall when I was a postdoc.)

Of course, you can also work on stuff related to your PhD topic, if it continues to lead to interesting research. But it's usually not good to restrict yourself to only doing this.

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  • I see, unfortunately I am not having much mentorship during the phd. I also am still struggling with restraining myself to focus entirely on research and instead learn new things, read papers etc which are not directly connected to the thesis problem, but may give good insight on new problems/techniques. I have the tendency to really stick to the phd topic's problem and kinda shut everything else off
    – Alex999
    Commented May 24 at 9:40
  • @Alex999 That's not the worst thing in the world when you're two years away from graduation. It's expected that PhD students will be very focused on their thesis problem. But now is a good time to start broadening your horizon gradually, by going to talks, having discussions, reading the abstracts of papers uploaded to arxiv in your field, etc.
    – user187020
    Commented May 24 at 18:14
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It seems to me that you don't have a birdseye view of where your work fits into your corner of mathematics. This is normal for a grad student, but you should start learning this relatively soon.

To this end, I think you need to attend more conferences and read more papers in your field. If your advisor isn't making you read stuff, I'd suggest starting with work of people you and your advisor cite frequently. (Also ask your advisor who the prominent people in your field are, especially ones who see things differently and/or write clearly.)

When you go to a conference in your field, don't just talk to other grad students, but tag along with the senior people in your field. It's very uncomfortable, but it's a great way to learn the "zoomed out" view of how your field is structured. I learned a huge amount of math you can't easily find in books or papers from uncomfortably hanging around during smoke and coffee breaks.

It's very common for a postdoc to attack a completely different problem from their thesis problem, using technical tools they became expert in during the PhD. Or they learn a new thing they can combine with an old thing. I can only think of a handful of examples of people who basically did a second PhD with no connection to the prior work.

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