## 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.