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I will soon finish my PhD in arithmetic geometry. My advisor told me that I will have to find my next research topic on my own. How do I do that? (Except for "continue where the PhD thesis ends") Can you give me hints/advice?

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closed as off topic by Misha, Mark Sapir, Todd Trimble, Vladimir Dotsenko, Felipe Voloch Jun 4 '13 at 22:38

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I don't understand. You've been doing research in a deep and wide field of mathematics, and your head isn't filled with questions and topics and ideas you're really itching to pursue once the headache of the PhD requirements is over and done? I like Ricky's "extraverted" advice, but me, I'd also introspect and think of all those juicy loose ends that must have cropped up and are left unresolved. Surely you can find something this way? – Todd Trimble Jun 4 '13 at 16:34
I would say, if you are interested in mathematics, you know about what you want to find out more. -- If not, why do you want to do mathematics at all? – Stefan Kohl Jun 4 '13 at 16:39
Although I agree with the earlier comments, I can see a possible point to the question. It's likely that the questions you know you want to pursue are ones that better and more experienced mathematicians have already attacked and probably are still attacking. So how do you choose something that you have a chance at solving but will be worth publishing in a good journal? – Deane Yang Jun 4 '13 at 17:11
Why is everyone so mean-spirited? Just some small words of encouragement and generic advice from more experienced colleagues and the OP would have gone on happily ever after :) More seriously isn't it very important to choose problems smartly to have a nice career rather than randomly investing time in the first thing that catches your fancy? – Eleanor Von Hohlandsbourg Jun 4 '13 at 22:35
The moral is that research doesn't have to begin with the idea "I want to prove such-and-such (new) result"; it may well begin with the idea "I want to clearly understand such-and-such (known) result." With luck, it will lead to new results, and even if it doesn't, you'll have learned something that can be useful later. – Andreas Blass Jun 5 '13 at 0:56

You should read literature in the area of your previous research, in search of interesting unsolved problems. Also attend conferences and talk to specialists in your field. When I was on this stage of my career, I found surveys with lists of unsolved problems in my field, and tried to solve them. Now I make such lists myself, to help young researchers, and I suppose they exist in every field of mathematics. You have to read a lot to become an expert in your area.

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