Texts: An Introduction to Topology and Homotopy, by Allan J. Sieradski, is the text for fall and a reference all year. The tentative text for winter and spring is William M. Boothby, An Introduction to Differentiable Manifolds.

Prerequisites: For 544, topology of metric spaces, basic facts of "naive set theory," elementary group theory and the usual "mathematical maturity to handle a graduate level course." Specifically, a student should either know or be capable of learning quickly by independent study the material in chapters 1; 2; 3, §1-2; and 11 of Sieradski. (Note: many of the ideas in chapters 2 and 3 will be covered in more general form in chapters 4-8.)
For 545, the prerequisites are 544 and the following topics:
(i) Vector calculus: partial derivatives, the total derivative as a linear map, the inverse and implicit function theorems, multiple integrals including change of variables, gradient, divergence, curl, and the theorems of Green, Stokes, and Gauss.
(ii) Ordinary differential equations: Existence and uniqueness of solutions and some facility for solving first order equations.
(iii) Linear algebra: abstract vector spaces, linear maps, change of basis, inner products, bilinear forms, linear functionals, dual spaces, and some aquaintance with canonical forms (e.g. Jordan canonical form).
For 546, the prerequisite is 545 (and thus all of the above).

Course content: Manifolds are generalizations of curves and surfaces; that is, they are spaces that locally look like Euclidean space (Rⁿ) but globally may not, just as the sphere locally looks like the plane. They are the basic subject matter of differential geometry, but also play a role in many other fields of mathematics, including algebraic topology and geometry, partial differential equations, and number theory. They are increasingly important in applications such as mathematical physics and computer graphics. Fall quarter will cover chapters 4-10 and 12-14 of Sieradski: the topology of manifolds, i.e., properties invariant under continuous deformation. The culminating concepts of this quarter will be the fundamental group and covering spaces; one application of the former will be the complete classification of closed (compact and without boundary) surfaces. Winter and spring will cover the material in chapters I-VI of Boothby: differentiable manifolds, tangent vectors and spaces, vector and tensor fields, ordinary and partial differential equations (in geometric form), differential forms and integration. Lie groups will be an important recurring example in these quarters.

Homework and grading: Grades will be based on homework and a final exam which may include both in-class and take-home work. Homework will be due about once a week.