In grad school, I’ve noticed that some people can actually get bored of math!
How does this happen?! I believe it often happens by not doing your own thing. Doing your own thing means pursuing problems that interest you, and learning whatever things that strike your fancy. It also means keeping your train of thought directed with your natural interests.
How do you do your own thing? It starts by avoiding things that you don’t care about. Here’s an example: Your advisor really likes a certain book and recommends that you read it. Should you? Only if you like it. If after a couple of chapters you find yourself not caring about any of the theorems, then all of a sudden you’re doing someone else’s thing. Perhaps the book is legendary, and there are legions of top graduate students who have been enlightened by it. Perhaps you think you can’t get through it because you’re not smart enough. However, if you don’t like it yourself, then there’s no point in struggling with it just to prove a point. Struggling isn’t always bad. Struggling is good if your quest is from the heart. If not, then you’re just eroding your natural enjoyment of life, which is hard to get back.
Doing your own thing means finding your own perspective and approach. It’s not good to blindly accept the approach of books, papers, or people. Say you want to learn about some big field, like the Langlands program. If you ask someone else, they might say “you should have a solid understanding of the structure of Lie groups first!” If you just listen to this advice without considering other options, you could read eight hundred pages of Lie groups without realising that there are concrete open problems in the Langlands program that require no knowledge of Lie groups to appreciate or even work on.
Doing your own thing is not contradictory to having an open mind. Mathematics that isn’t interesting now may become interesting with the right approach and a little time. In fact, I believe there is no boring math, only approaches that are incongruous with oneself. For me, an excellent example is algebraic geometry. When I first saw algebraic geometry, I was not impressed. It’s only when I saw the applications of algebraic geometry to representation theory via the fundamental lemma that I actually became interested in algebraic geometry. So the motivation has to be right, and sometimes it takes much time to find that motivation. You should know what you like, but be ready to like new things as well.
Throughout my PhD, I’ve managed to keep reading books and papers of my choosing and working on whatever problems I like, and I suggest you to do the same. In short, I hope I’ve convinced you to do your own thing. Listen openly to the ideas of others but in the end, just do what you like!