I'm about to start my first year of studying mathematics. My university requires me to take quite a lot of basic courses in the first year and after that there are some strict limits on the number of courses I can take. This means that I will not be able to take all the courses I would like to before graduating. I have done some self-study prior to enrolling and believe that I could fairly easily skip some of the early courses and I know that the university allows this, but I'm unsure how it will be looked upon by graduate schools. If for instance I took "Algebraic topology 2" (second graduate algebraic topology course) and got an A would it be assumed that I know the content of topology and "algebraic topology 1" even if I didn't take them? Would such an approach present a problem if I have made sure I can handle the course? (for instance by auditing some algebraic topology 1 lectures earlier or simply checking the prerequisites and earlier lecture notes) In addition I know that there is an open lecture policy in place which basically means that I can attend any lecture I want, and since no attendance is kept in any courses I'm thinking of skipping some easy lectures like "intro to analysis" (which is a required course) and go see some more advanced course like commutative algebra, but I'm somewhat worried since this won't show up on my transcript so unless it's part of a sequence like "analysis 1 (real) -> analysis 2 (complex) -> analysis 3 (functional)" where I plan to take the last courses in the sequence, then I won't really have anything to show for it. Is this approach to studying a bad idea? I guess I'm just a bit concerned because it feels like I have to choose between optimal learning and what looks better to graduate schools. Am I simply a bit paranoid about how the system works or are these concerns valid? I would love to hear some other people's perspective on this situation.