Mathematics 790-90-02, Spring 2015

Mathematics 790, Spring 2015

Minicourse: Morse theory and Floer homology

Time: Wednesdays and Fridays, 10:05-11:20am
Dates: March 18 to April 15
Location: Physics 227

Lecture notes (in progress; I'll update these when I can)

Morse theory has historically served as an important intermediary between differential geometry and algebraic topology, and it has gained further importance in topology in the last few decades through the work of Floer, Witten, and many others. This minicourse will provide a rapid introduction to Morse theory with the goal of defining Morse homology. We'll then pivot to Floer homology, which can be thought of as an infinite-dimensional analogue of Morse homology, and discuss what's different in this setting.

Topics I plan to discuss:

Morse theory: Morse functions, surgery and handles, Morse inequalities, Morse-Smale pairs and gradient-like flows
Morse homology: the Morse-Smale-Witten complex, orientations and signs, Poincaré duality
Floer homology: an overview of selected flavors, selected from Hamiltonian Floer homology, Lagrangian intersection Floer homology, Seiberg-Witten Floer homology, and Heegaard Floer homology. (This is probably wildly optimistic for a minicourse, but we'll see.)

I'll assume a basic familiarity with algebraic topology along the lines of Math 611 (people taking 611 concurrently should be fine). It may also be helpful to be comfortable with differential geometry along the lines of Math 633, but this isn't so necessary.

If you'd like further reading material, there are now many good books on Morse theory and Morse homology, some with discussions of Floer theory as well. These include:

Liviu Nicolaescu, An Invitation to Morse Theory
Michèle Audin and Mihai Damian, Morse Theory and Floer Homology
Augustin Banyaga and David Hurtubise, Lectures on Morse Homology
John Milnor, Morse Theory (a classic, but doesn't cover recent developments).

The first two have the added virtue of being electronically available through the Duke Libraries web site.