I took my qualifying (oral) exam in Fall 2019. My committee members were Mark McLean, John Morgan, and Olga Plamenevskaya. Here is a document with questions I thought would be fair to ask as well as a transcript of the actual exam. Below are some notes I wrote up to help me study. Please let me know if you spot errors.

The main source for the major topic was *Morse Theory and Floer Homology *by M. Audin
and M. Damian. It is an excellent book and so my notes have little to add
to their treatment. Perhaps the main downside of the text is that it's
fairly lengthy. Hence, in preparing for the exam, I found it helpful to
condense material into these notes.

- Definitions and equations
- Oriented Morse Theory
- Morse Invariance
- Morse Theory for Closed 1-Forms
- A brief story of Hamiltonian Floer theory and one of Arnold's conjectures
- Periodic orbits
- The action functional
- Ch. 6 notes: Compactness of the Hamiltonian Floer moduli spaces
- Ch. 7 notes: Conley-Zehnder Index
- Ch.
8 notes: Transversality

- Ch.
9 notes: Gluing

- Ch.
10 notes: Floer to Morse

- Ch.
11 notes: Invariance

For the minor topic, I mostly referred to John Morgan's *The Seiberg-Witten Equations and Applications to the
Topology of Smooth Four-Manifolds. *It was also helpful to use
his notes in *Gauge Theory and the Topology of Four-Manifolds* and
Lawson and Michelsohn's *Spin Geometry. *As above, all these
sources are already excellent and my goal was mainly to condense them.

- A not-so-brief overview of Seiberg-Witten theory
- Short note on Principle G-bundles
- Short note on spin geometry
- Almost complex 4-manifolds in Seiberg-Witten theory
- Seiberg-Witten invariants of some Kahler surfaces

Perhaps because of the pressure of the exam, my mind cracked a bit and I
wrote something entitled: "If
Complex Geometry Became a Gladitorial Sport."