Marking Scheme:
The course mark will be based both on (1) attendence (2) an essay or
presentation.
The deadline for "essays" is Monday, March 29. The "presentations" will
take place after that.
The essay should be on some special topic related to Morse theory or
applications of Morse theory, and approximately 3 pages in
length. Usually such an essay shouldn't contain detailed proofs, but
rather explain the main ideas or applications. In place of a written
essay, you can also give a short presentation on such a topic
(approximately 20-25 minutes) -- either in class or in my office.
Below are a couple of suggested topics -- if you have your own idea (it doesn'thave to be all that fancy!), please let me know.
1. The Morse-Witten complex (Witten, J. Differential Geom.
17 (1982) 661--692). See also M. Guest for a survey and
additional references.)
2. Moment maps as Morse functions. See e.g. Guillemin-Sternberg's book
"Symplectic techniques in physics", or Kirwan's book, or
Tolman-Weitsman,
on the cohomology rings of Hamiltonian T-spaces.
3. Circle-valued Morse theory (Faber, Ranicky, Hutchings, ..)
4. Holomorphic Morse inequalities (E. Witten, J.-P. Demailly,
Siye Wu, M. Braverman, ..)
5. Morse theory on graphs: (cf.
Guillemin-Zara and later paper by these authors.)
6. Morse index and Maslov index: E.g. Duistermaat, "Morse index in
variational calculus", Advances
in Mathematics 21 (1976), 173-195
7. Morse-Smale functions.
8. Harvey-Lawson theory. See
Harvey-Lawson, "Finite volume flows and Morse theory",
Annals of Math. vol. 153 no. 1 (2001), 1-25,
Harvey-Lawson, "Morse theory and Stokes' theorem".
Both papers are available at
Lawson's webiste
For most of these topics, you should be able to find papers in the
archive.
Also, Martin Guest's paper
Morse theory in the 1990's contains a lot of references.
Or else, just ask me!