stingc

A miniscule fraction of modern mathematics research is ever applied to 'practical' problems. Mathematicians largely do what they do for their own amusement. However, their work is still viewed as "applicable" by their peers (at least by the small handful who might understand it). A result is considered applicable to them if it is interesting and leads to more interesting problems. "Interesting" here basically means surprising and/or beautiful, not practical in any technological sense.

Applied mathematicians are usually interested in physics, but there is still a big gap between those two fields. Surprisingly enough, physicists hardly talk to the applied math people, which is mostly because they're too hard to understand. The math people don't really make an effort to change this either, as they're usually concerned with problems that physicists don't consider interesting (e.g. existence proofs).

In general, most mathematicians seem to write without much regard for how their work will be received by others. They will often spend a very large amount of time making sure something is perfect, and then write it all up more as a way of preserving the knowledge they have discovered rather than teaching it. Most of the sciences are much more competitive. Perfectionism is rare, and usually leads to failed careers because it takes too long to publish perfect papers. Peer acceptance is also a much bigger priority.

Anyways, I suppose the more practical sciences are affected somewhat by popular culture, but there are branches where I don't think there is any connection at all.

I don't know if you're trying to limit this discussion to popular culture though. Math and science both have their own internal cultures, and these very strongly influence which problems people choose, and how those problems are approached. These cultures are often very specialized. In my own field, you can often tell who someone's Ph.D. advisor was by looking at the style of their papers.