I was a math + physics major in college, and it was more than 3 decades ago, so take this with appropriate grains of salt. What I would have called high level math in the curriculum that I studied, wasn't so much about specific topics, but about the sophistication and creativity of your approach.

The engineering / science math was pretty much a matter of looking at the problem, guessing its "form," and applying a known technique based on that form. For instance, "this looks like integration by parts." Eventually you'll be shoved out into the world where there are problems for which there is no known solution, and you have to create your own techniques.

The more advanced courses did two things. First, they set aside problems and advanced towards proofs. This is really where math came alive for me. Proofs are so much more varied that you have to abandon the security of a bag full of known tricks. The other thing is that the derivations get longer, so you have to develop a longer train of thought, if you will.

Courses that were higher-level were like:

abstract algebra

complex analysis

real analysis

topology

I don't know what new goodies there are, but I'd love to dive back into it again.