
Math 8061: Smooth ManifoldsFall Semester 2010
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 R^{n}? 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. Textbooks: I plan to draw material from two books:
Prerequisites: Concepts of analysis (Math 504142) and abstract algebra (Math 8011). The algebra course is more of a corequisite, as we will not need much algebraic material until the second semester.
Grading Scheme
Homework policy: Homework assignments will be posted on the course webpage, and will typically be due on Thursdays. 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 takehome final will be handed out during the last week of classes, and will be due on December 15. dfuter at temple edu Last modified: Fri Aug 21 13:41:22 PDT 2009 