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?
closed as off topic by Misha, Mark Sapir, Todd Trimble♦, Vladimir Dotsenko, Felipe Voloch Jun 4 '13 at 22:38
Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question.
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.