Manifolds are the generalizations to arbitrary dimension of curves and surfaces -- spaces that locally look like Euclidean space, just as the sphere locally looks like the plane. They are the basic subject matter of differential geometry, but also are increasingly useful in many other branches of pure and applied mathematics. For more discussion of what manifolds are and what they are good for, see the introductory chapter of the 544 text, [ITM].

In Autumn quarter, we will focus on the topology of manifolds, that is, the properties that are invariant under continuous transformations. This serves as an excuse to study topology in somewhat greater generality, not just on manifolds. Although we will start with basic definitions, students are assumed to have prior experience with topology in the context of metric spaces (for instance, in a senior level real analysis course). Major goals include understanding the fundamental group, covering spaces, and the classification of compact surfaces.

Topics in 545/546 will include differentiable manifolds, vector and covector fields, flows, the Frobenius theorem, Lie groups, homogeneous spaces, tensor fields, differential forms, Stokes's theorem, and deRham cohomology.

Texts: Introduction to Topological Manifolds [ITM], by J. M. Lee is the text for 544. Introduction to Smooth Manifolds [ISM], by J. M. Lee is the text for 545/6.

Prerequisites: As you can tell from the list of topics below, the study of manifolds draws on a variety of areas of mathematics that are (hopefully) part of your previous education. It is not necessary to have complete command of all the topics listed here. But if any are completely missing from your background, or your skills in several have faded, you probably will find this course even more challenging than some of our other first year grad courses. In particular, you probably should not take Manifolds if you have such gaps in your background and this is your first year of graduate school. See me if you have questions about your preparation for the course.

For Math 544, prerequisites include the following (all reviewed in the appendix of [ITM]).

• Basic set theory.
• Topology of metric spaces: Metrics (distance functions), open and closed sets, convergence and continuity, compactness.
• Elementary group theory: homomorphisms, isomorphisms, subgroups, normal subgroups, cosets and quotient groups, cylic and permutation groups.

• Linear algebra and calculus: the material reviewed in the appendix of [ISM]. Prior acquaintance with the Inverse and Implicit Function Theorems and vector calculus (divergence, curl, the theorems of Green, Stokes, and Gauss) will be helpful.
• Differential equations: basic facts about existence and uniqueness of solutions to ODEs; elementary techniques for solving first-order equations and systems.
• Math 545 is prerequisite for 546.

Grades: will be based on weekly homework and a take-home final exam. Homework will include a brief reading report by e-mail as well as (hard copy) problem solutions.