Geometry of Manifolds (Math 966) Spring 2013

Larry Guth
Office: 2-371

Class times: MWF 2-3, 2-146.

Description of the class: The theme of the class is the connection between analysis on the one hand and the topology of manifolds on the other hand.  There are three main topics.  The first is transversality, covering Sard's theorem and its applications in topology.  The applications include degrees of maps, linking numbers, and the Hopf invariant.  The second topic is vector bundles, connections, and characteristic classes.  We will study the Euler class, and Chern and Pontryagin classes.  One of the main results we will study is the Gauss-Bonnet-Chern theorem.  The third topic is Morse theory, connecting the critical points of a function to the topology of the manifold.  We will begin with Morse theory on finite dimensional manifolds, and then study Morse theory on the space of loops on a manifold, building up to a proof of the Bott periodicity theorem on the homotopy groups of the unitary group.

Texts: In the first unit, we will use the book Topology from the Differentiable Viewpoint, by John Milnor.  In the third unit, we will use the book Morse Theory, also by John Milnor.  For the second unit, we will use in-class lectures, and I may post some references on the webpage.

Homework and grading: We will have 6-7 problem sets during the course (about one every two weeks).  I will drop the lowest grade and average the other grades.  (This means you can miss one homework assignment.)  You are encouraged to work together on the problem sets, but you should write up your own solutions.

Class Announcements:

The week March 11-15, I will be out of town.  On Monday March 11, there will be an in-class problem/activity.  On March 13 and March 15, there will be guest lectures by Tom Mrowka.

End of the semester.  The last problem set will be due May 10.  The last day of class will be May 15.  On May 17 at 3 pm, I will have office hours.  I'll be happy to talk about questions from the class, or about differential geometry more generally, ...

Problem sets and assignments:

Letter of introduction This is a letter about your interests and background.  The file in the link has some particular questions that will be helpful for me.  Please email me sometime this week (Feb. 11-15).  (If you already wrote me, you don't have to write again, but feel free to add something if you would like.)

Problem Set 1
Due on Friday, February 22 in class.

Problem Set 2 Due on Friday, March 8 in class.

Problem Set 3 Due on Monday, April 1 in class.

Problem Set 4
Due on Friday, April 12 in class.

Problem Set 5 This is the last problem set of the course.  It will be due on Friday, May 10 in class.