Applied math programs rarely make you so Diff geo, topology. From what I have seen
Analysis ( advanced calculus )
Proof based Linear Algebra
Electives in Diff Geo, topology, ect.
Differential equations + Partial
Numerical analysis ( computer math programming)
Elective math course.
These are the norms at most university's. I am not sure what math class you have taken thus far, but upper division pure math will be far from anything you have done before. I am only in calculus 1 but have self studied a lot of pure math with a professor and it is SO much different and much more interesting to me since I hate application problems.