30% homework, 20% oral presentation, 50% final paper.
Problem set 1 is here.
Problem set 2 is here.
Problem set 3 is here.
Problem set 4 is here.
Problem set 5 is here.
Morse theory is an extremely simple tool which has revolutionized fields of mathematics several times over. Morse himself developed the theory and applied it to mathematical physics. Later, Bott took these ideas and used them to prove his celebrated periodicity theorem. Then Smale used it to prove the h-cobordism theorem, which implies the generalized Poincare conjecture in dimensions five and above. More recently Andreas Floer applied the ideas in the symplectic setting to prove the Arnol'd conjecture, and in the process invented an important new homology theory.
This tutorial, however, will have the goal of introducing the basic ideas and proving Bott's periodicity theorem. This theorem is concerned with the topology of matrix groups, and demonstrates a very beautiful and suprising fact about their higher homotopy groups. In a larger sense, though, another goal for the course will be to play with manifolds and the tools that we use to understand them. We will get to talk about Lie groups and differential geometry (two extremely important parts of mathematics) in very elementary, hands-on ways. It should be a lot of fun!