It seems like every other graduate student I know is working on some problem that their advisor has suggested, and I've just been spending a year reading papers and trying to think of a problem for myself. Is something wrong here? Are you supposed to come up with a thesis problem on your own?
I am only looking for answers which are specific to pure math. I have friends in compsci and various sciences, and their research lives are not similar to mine at all.
I am interested in an area of math which has a similar flavor to the one that my advisor works in, but is quite different. My advisor thinks that this area is a good one to work in and knows of some of the major results, but doesn't know any of the details of the proofs or what people are working on now. I am finding it very hard to come up with good problems on my own. I can come up with ideas for problems, but I cannot judge whether they are interesting to other people who work in this area or feasible at all. I also can't tell for sure whether something has been done before, as I have to rely on mathscinet since I am not an expert in my field and do not have experts to talk to. I think these are the things that an advisor should be able to help with, but right now I feel like I'm just all on my own, even though I meet with my advisor frequently to check in. I also feel like I'm getting behind in graduate school because I still haven't found a single problem to work on after two years.
There's nothing necessarily wrong. There are different ways to a thesis. Many advisors expect students to come up with their own problems and others don't. For certain fields/people this is easier than others, but in general finding good problems is hard and takes time. Choosing good problems to work on is a large part of being a good mathematician. Often experts who give problems to students will consult with other experts before giving a thesis problem or before deciding on whether to work on something themselves.
There are many successful mathematicians who wanted to do something in a different area than their advisor. However, it is a bit riskier, as you seem to realize, for several reasons: it may be hard to find a problem, you want to make sure what you do isn't already known, you may want some perspective/guidance, you want to make sure what you do is interesting, ...
Here are some things you can consider, in no special order:
Look for a secondary advisor. You should of course ask your advisor what he thinks about this, and he may be able to suggest some people.
Go to conferences/workshops/etc in this area (also local seminars if possible) and meet people and talk to them about ideas. You shouldn't expect someone to give you a problem, but if you have some general ideas for things to try, they may have valuable specific suggestions.
See if you can visit an expert at another institution. Again, talk to your advisor about this and perhaps he can make suggestions.
If you know how to do some things, you can just do them and write them up, then send them to experts and ask for feedback (is this known? is it interesting?).
Do your thesis in your supervisor's area, and then you can consider doing work in this other area later.
At the minimum, talk to your advisor about your concerns about feeling "lost." That's one of the things they're there for.