What do US universities mean when they mention anything above the "Calculus" course?
I think your question is: What does calculus mean?
Calculus would usually include learning to compute derivatives and integrals.
If you are learning to prove the theorems used to compute derivatives and integrals, that would be more advanced than calculus.
linear algebra analysis 1, multivariable analysis, topology, group theory
All of those would usually be considered more advanced than calculus.
This probably means anything beyond the semi-standarized three introductory Calculus courses.
Examples include:
- Differential Equations
- Linear Algebra
- Discrete Mathematics
- Probability
- Statistics
- Ring Theory
Or basically topics that might consider Calculus as a prerequisite to performing well in the class.
Classes that build a mathematical foundation to take calculus won't apply, like:
- Algebra
- College Algebra
- Pre-Algebra
- Any math topic "for some other non-math major" (Statistics for Business majors)
To clarify I've attached the University of Houston's Math department class offerings. Note that Calculus I, II, and III are 1000 (Freshman) and 2000 (Sophmore) level courses. I would assume any 3000 or 4000 level course would satisfy the requirement, and possibly some of the 2000 level courses.
This should help clarify some of the comments about "Algebra" courses. Higher level Algebras that would be post-Calculus include:
- Elements of Algebra and Number Theory
- Abstract Algebra
I hope this provides a little more clarity.
As far as the US is concerned, "Calculus" is the first introduction to the material. It typically is light on proofs and often geared to the Engineering Curriculum. In Germany (where I grew up) this material was partially high school, partially (in College) classes called ``Higher Mathematics for Engineers''.
To get a more detailed idea, http://www.cds.caltech.edu/~marsden/volume/Calculus/ are (by now somewhat old, but the material has not changed) Calculus textbooks that would be at the upper (more ambitious) level -- many books are weaker.