It's coming down to the wire. You've finished your classwork; the breadth requirement is out of the way. You passed your qualifying exam. Only one thing stands between you and graduation: the thesis!
But now...how do you choose a good
PhD thesis topic? It seems as if this is something you should know; after all, you've been
going to grad school for years. But somehow, it seems you never took a seminar on "how to choose a thesis topic", and suddenly, here you are without one!
Choosing a thesis topic is extremely important, as the wrong decision can easily keep you from getting your degree, or at least prevent you from finishing on time. If you're lucky, your committee will reject your topic immediately, sending you back to the drawing board. If you're unlucky, you may spend several years working before realizing that solving the problem is beyond your current abilities.
So how do you choose a good topic? First off, you should be at least somewhat familiar with the area. You just spent several years taking classes; now isn't the time to jump to another area that you know nothing about! Pick something you've been studying and are interested in and read more in that field; get a sense of what's known and what's not known. Find a list of open problems and see if your reading suggests a possible approach to one of them; old problems are often solved by applying known techniques that have never been used for that particular problem before. Once you have a possible problem, take it to your advisor and ask his opinion! He should be able to give you an idea of whether the problem is solvable within a reasonable amount of time. He can also tell you if he doesn't think it's significant; if the committee is going to reject the problem anyway, much better to find that out before you devote much time to writing the proposal!
Once you get your topic picked out, you'll need to write up the proposal, which should tell your committee why the problem is important, demonstrate your background knowledge in the area, and explain how you plan to solve it. Good luck!
Education 
