Real Analysis: A senior level year-long course in real analysis, including basic set theory and metric topology as reviewed in the appendices A and B of the 544 text [ITM]. In 545/6, analysis in Rn, including differentiation, integration, Taylor's theorem, uniform convergence, and the inverse and implicit function theorems, and basics of elementary differential equations, as reviewed in Appendices C and D of the 545/6 text, Lee's Introduction to Smooth Manifolds [ISM].
Algebra: Group theory as reviewed in appendix C of [ITM]. In 545/6, linear algebra plays a surprisingly large role; the needed facts are reviewed in Appendix B of [ISM].
Vector Calculus. In 545/6, we generalize the ideas of gradient, divergence, and curl, and the theorems of Green, Stokes, and Gauss. (Gauss's theorem is also known as the divergence theorem.) While the vector calculus versions are not used directly, students who remember them well usually find the generalized versions a bit easier to learn and work with.
Many (not all) first year math graduate students find the Manifolds sequence a bit harder than the Algebra, Real Analysis, and Complex Analysis sequences. The next paragraph has my best guesses as to the reasons why this is so. If you are debating which courses to take, consider this information, and feel free to come discuss it with me further.
Math 504, 524, and 534 are somewhat like continuation of the corresponding undergraduate courses. Manifolds builds on ideas you've seen previously in several different math courses. This means when you're stuck on a problem, it may be less obvious where to look for the tools needed. The review of point set topology (Chapters 2-4 of [ITM]) is very rapid, and then we start into topics where some students feel the style of proof is unlike their previous math courses. They even sometimes complain they don't know what it would mean to prove some types of mathematical statements. (There are lots of good models for proofs in the book, but some of the techniques take a bit of getting used to.) In 545/6, some of the prerequisites (see above) are distant memories.
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