>>13001325Ideally by the time you finish your Bachelor's, you will have taken at the senior undergraduate level (or equivalently basic graduate level) 2 courses in real analysis (measure theory, functional analysis) and 2 in algebra that cover at least galois theory and commutative algebra. Hopefully also a point set topology course and maybe something else.
Then you do your Master's and take a bunch of courses doing various things like complex analysis, algebraic topology, differential geometry, algebraic geometry, probability theory, differential equations, whatever amount of breadth needed to satisfy degree requirements.
Only the strongest of students graduate ready to jump straight into a PhD program. American college first-year is often a complete joke, like an overgrown 13th year of high school, to filter out all the clowns (and diversity admissions) who have no business being there. Even second year at many schools can be laughably tame, so often you see math programs don't really get going until the third year (where basic competence in proofs and rigor are expected).
