For an overview of what manifolds are and what they are good for, see the introduction to the fall quarter text.

Text:  For the fall, Introduction to Topological Manifolds by John M. Lee. For the winter and spring, Introduction to Smooth Manifolds, also by John M. Lee. (The latter text is not yet available, but will be at the appointed time.)

Prerequisites:
For 544:

• Set Theory: Basic facts of "naive set theory'': properties of the real numbers and euclidean space, functions, equivalence relations, countability, order relations, the well-ordering theorem.
• Algebra: Elementary group theory: homomorphisms, isomorphisms, subgroups, normal subgroups, cosets, quotient groups.
• Analysis: Open and closed sets, continuous maps, convergence, metric spaces, compact and connected sets.

• Vector calculus: Calculus of mappings from $R^n$ to $R^m$, partial derivatives, the chain rule, the total derivative as a linear map, multiple integrals, the change of variables formula, the inverse function theorem, divergence, gradient, curl, and the theorems of Green, Gauss, and Stokes.
• Ordinary differential equations: basic theory and techniques at the level of Math 307 and 309.
• Linear algebra: abstract vector spaces, dimension, linear maps, change of basis, Jordan canonical form, inner products, bilinear forms, linear functionals, dual spaces.