Given that pure mathematics is, by definition, not concerned with applications, how does one decide that one problem is more valuable than another? Is it just a matter of certain topics becoming popular among the community of research mathematicians? Is it just a matter of doing something very difficult? Is it about solving something that people previously failed to solve? Is it about doing something that will get you a better job, and if so, how do employers decide what is "important?"
I know from experience that when one is working on an interesting problem, this question tends to matter less. The math is interesting for its own sake. But that doesn't mean it is going to be deemed valuable by the greater community. I also feel that much of mathematics, especially at the beginning of one's career, is motivated by trying to impress people who are more established. But how do they decide which problems matter and which ones don't? Is it just a matter of taste?
I expect that this question might be closed or voted as off-topic, etc., but I feel strongly that this is a valuable question for a mathematician to have an answer to. It's also something one is always confronted by as a teacher -- one needs to have an answer when students inevitably ask "Why do we need to learn this?" Also, being someone who has completed my PhD and has a (so far) successful research career, the fact that I don't have a totally solid answer to the question evidences its non-triviality.