Course outline: This course will be an introduction to the geometry and topology of smooth manifolds. We will begin the fall semester with the definitions: what does it mean for a space to (smoothly) look just like Rⁿ? We will go on to study vector fields, differential forms (a way to take derivatives and integrals on a manifold), and Riemannian metrics. In the spring semester, we'll study the interplay between the geometry of a manifold and certain ideas from algebraic topology. We will review the idea of the fundamental group and introduce homology - and then relate these algebraic notions to the underlying geometry. If time permits, we will talk a bit about hyperbolic manifolds - a family of manifolds where the interplay between topology and geometry is particularly strong and beautiful.

Textbook: Introduction to Smooth Manifolds by John Lee.

We will cover parts of Chapters 1-8, 11-14, and 17. Those chapters will likely seem like too much material for one semester, and they are. I believe Lee's book is very well written, and an excellent reference, but is too encyclopedic to be covered in a linear fashion. We will bounce around to some degree.

To get a better intuitive sense of the topics that we'll cover, as well as the order in which we'll cover them, it helps to look at Chapters 1-8 of A Comprehensive Introduction to Differential Geometry (volume 1) by Michael Spivak. The latter book is not required, but makes a pretty good companion source.

Prerequisites: Concepts of analysis (Math 5041-42) and abstract algebra (Math 8011). The algebra course is more of a co-requisite, as we will not need much algebraic material until the second semester.

Homework policy: Homework assignments will be posted on the course webpage, and will typically be due on Wednessdays. No late homework will be accepted, but I will drop your lowest homework score. I encourage you to start early and to discuss the problems with other students. By all means come by my office hours if you have trouble with a problem. The only real caveat to group work is that you must write up your own solutions, in your own words.

Final Exam: The take-home final will be handed out during the last week of classes, and will be due on December 12.