Course Schedule

The following is an approximate outline of the course. After each lecture I will update it with more details on what we covered and so the appromximate outline will become exact. I will try and stick to the plan below, however the predictions for future lectures may change depending on how fast or slow we go.

Date	Topics covered	Recommended Reading
Friday August 16	Class Overview Morse functions, the Morse Lemma Some background on manifolds and vector fields	Milnor, Chapters 1 and 2.
Friday August 23	Finish proof of the Morse Lemma Critical points of a Morse function correspond to attaching cells Background on manifold theory: Metrics, the gradient vector and flows of vector fields	Milnor, Chapters 2 and 3. A good reference for the material about flows on manifolds is: Jost, Riemannian Geometry and Geometric Analysis, 4th ed.
Friday August 30	Finish proof of cell attachment theorem Some applications The Morse inequalities Some applications of the Morse inequalities	Milnor, Chapters 4 and 5. Click here for a visualisation of the three critical point torus example. Another cool application of Morse theory (cohomology of flag varieties)
Friday September 6	More on Morse functions for projective space The idea behind Smale's proof of the Poincare conjecture Existence of Morse functions Background on smooth vector bundles	Milnor, Chapter 6. Chapter 1 of Hatcher's book is a good reference for the basics of vector bundles
Friday September 13	Existence of Morse functions eLearning Week Instead of a lecture in class, I will post a video lecture on IVLE.	Milnor, Chapter 6.
Friday September 20	The Lefschetz hyperplane theorem The Thom isomorphism and the Euler class	Milnor, Chapter 7. Good references for the Thom isomorphism are: Milnor, Characteristic Classes (Chapters 9 and 10) Hatcher, Algebraic Topology (Section 4.D) Hatcher, Vector bundles and K-theory (Chapter 3)
Friday September 27	NO CLASS: MID-SEMESTER BREAK
Friday October 4	Assignment 1 due Introduction to Morse-Bott theory. The Thom isomorphism and the Euler class.	Bott, Nondegenerate critical manifolds.
Friday October 11	The Morse complex.	Austin, Braam, Morse-Bott theory and equivariant cohomology.
Friday October 18	The Morse-Bott complex and examples. Equivariant Morse theory
Friday October 25	Equivariant Morse theory (cont.) Cohomology of symplectic quotients	Kirwan, Cohomology of quotients in symplectic and algebraic geometry
Friday November 1	The Yang-Mills equations over Riemann surfaces	Atiyah, Bott, The Yang-Mills equations over Riemann surfaces
Friday November 8	Assignment 2 due Applications of Morse theory to the Dirichlet problem
Friday November 15	Class Presentations

Last updated 18 October, 2013