Suggestions for math post doc topic after PhD

Question

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.

Answer 1

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.

Answer 2

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.