Yes.

When I was a kid I wanted to get a PhD in math and physics and comp sci and etc etc etc. I was really disappointed when I got older and realized I would only have time for one. So I chose math because I wanted to study mathematical logic, differential equations, numerics and etc etc etc. Then I got to grad school and realized I would only have time for one. So I chose numerics because I wanted to study molecular dynamics, computational neuroscience, fluid dynamics, etc etc. Then I had to choose an advisor, and I realized I'd only get in an education in one sub discipline. So I chose fluid mechanics, because I wanted to study polymeric fluids, multiscale dynamics, fluid-structure interactions, etc etc. Then I finally started doing real research and realized I'd be lucky just to make a noticeable contribution to one of those.

There are exceptions, of course.

(I don't mean to sound pessimistic, just realistic. Research has been good to me, and I love doing it)

How often are you exposed, even minimally, to other disciplines inside mathematics?

For example, I'm no researcher, but I tend to try and read a few good books about maths, and sometimes even actual mathematical books, just for the fun. Do you tend to do the same?

It's really a big gap though between having an understanding of the basics in an area and really knowing enough to follow the current research. It wouldn't surprise me at all if this guy actually had looked at things in basically the same direction as some of this research and still isn't able to really follow what they have done.

Absolutely. I read up on a lot of science, and even chug through text books. As robryan says above though, reading a book on chemistry or functional analysis is long way from being ready to do research in the field.

One of the reasons I chose applied math was because I thought it was the best centralized position in all of science. It would take me years to switch to another branch of math, but in applied math I can at least switch the focus of application, possibly in under a year or two.

Oh, hell yes.