## Math 544/5/6, 2013-14: Prerequisites and Challenges

### Prerequisites

*Real Analysis:* A senior level year-long course in real analysis,
including basic set theory and metric topology as review in the appendides
A and B of our text [ITM]. In 545/6, analysis in **R**^{n},
including differentiation, integration, Taylor's theorem, uniform convergence,
and the inverse and implicit function theorems, and basics of elementry
differential equations, as reviewed in Appendicies C and D of the 545/6 text,
Lee's *Introduction to Smooth Manifolds* [ISM].

*Algebra:* Group theory as review 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.

### Challenges

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 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 geting used to.)
In 545/6, some of the prerequisites (see above) are distant memories.

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Math 544 Homepage.

* Most recently updated on September 24, 2013.*