What is it like to do mathematical research?
For me, it was very painful, so I quit. Most of the pain was due to my not being interested in the subject, but having to do it anyway because otherwise, I would have had to face the shame of spending 6 or 7 years in grad school with only a masters to show for it. I thought that I was interested in it, so I was lured into doing a dissertation on it. It's hard to appreciate just how obscure the stuff people are studying these days is, until you are in the thick of it. Everything is deceptively simple when you are an undergrad. The classical theories you study at that stage have something more compelling driving them. I got the feeling when I was doing my dissertation, that I was adding one tiny room to a labyrinth that was already full of thousands of rooms. It can make you feel pretty small, when you start to see the vastness of the mathematical landscape spread out before you.
I suggest not worrying about what problems are open if you want to take a crack at some practice research. I think that's a good idea. At one point in time, whatever you are working presumably would have been open. Probably, you won't be making a very significant contribution until well into your career. It's all mostly just an exercise at this point, even if you go to grad school. I have worked out several things that I know must have been figured out by someone else long ago, and that was the most fun I've ever had doing "research". Stuff I did because I wanted to, not because I had to. To be successful, it helps to have what you want to do coincide with what you have to do. It's not as easy as you might think to make that happen.
Given the difficulty, if you want to get to that point, I think it's a very good idea to start experimenting with research-like activities at an early stage. There aren't really any rules. You can either ask questions and try to answer them or you can just play around with stuff. Notice things. Sometimes, when you study different subjects, you get curious about whether there's a relationship between this and that because they seem so similar, and so on. You can ask questions like that and try to answer them.
Work on writing stuff up.
But be warned, it's not for everyone. I thought I was the biggest math nerd on the planet before I went to grad school, but even for me, the current research ended up looking extremely obscure. Very complex, and more in a sort of baroque, over-wrought sort of way than in a deep, beautiful sort of way. There's a lot of it that I find just plain boring and ugly. And there's quite a bit that I can definitely see the beauty and appeal of, but it just goes on and on and on, and at some point, it's just not really doing that much for me anymore. It's just not that novel to me. I've seen 1000 beautiful proofs. What does the 1001st add? Not too much, unless there's something really novel about it that I haven't gotten from the other 1000. But, as I always say, I'm a physicist at heart, so in the end, I always wanted to connect back to reality and more practical things, so maybe I was barking down the wrong tree all along. I mean, reality is so complicated and important. Why waste time studying unreal things that tell you nothing about it? Not to say that some of it won't eventually have something to say about reality, but until I see the relevance, for the most part, I have lost interest.
