Math 8061: Smooth Manifolds

Fall Semester 2010

Meets: Tue/Thu 2:00-3:20 PM in Wachman Hall, room 447
Instructor: David Futer
Office: 430 Wachman Hall
Office Hours: Tue 10:30-12:00, Wed 1:30-3:00, or by appointment
E-mail: dfuter at temple.edu
Phone: (215) 204-7854
TA: Michael Dobbins
Office: 523 Wachman Hall
Office Hours: TBA
E-mail: mdobbi01 at temple.edu
Phone: (215) 204-6769

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.

Textbooks: I plan to draw material from two books:

The two books cover similar material, but have very different styles. Spivak is very conversational and does a great job of providing intuition. On the other hand, sometimes it's hard to find a particular definition/lemma/theorem when you need it. Lee is a much more traditional definition/lemma/theorem book, quite well-written but perhaps overly encyclopedic. I plan to follow the trajectory of Spivak a bit more closely. On the other hand, Lee is a better reference.

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.

Grading Scheme

The main components of the final grade are homework and a take-home final exam.

Component Date Worth
Homework Thursdays 60%
Final Exam December 15 40%

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 take-home final will be handed out during the last week of classes, and will be due on December 15.

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Last modified: Fri Aug 21 13:41:22 PDT 2009