Here are summaries of each lecture. The references refer to the texts described in the syllabus.

10/2: Introduction. What is Morse theory about? The torus example. Non degenerate critical points. Morse’s Lemma. References: Hutchings, Section 1; Milnor, Sections 1-2.

10/4: Proof of Morse’s Lemma. Examples of critical points. Flows of vector fields. Homotopy type in terms of critical values, part 1: “homotopy type stays the same if we do not cross a critical value”. Reference: Milnor, Sections 2-3.

10/9: Homotopy type in terms of critical values, part 2: what happens when we do cross a critical value? Reference: Milnor, Section 3.

10/11: Reeb’s theorem on nondegenerate functions with two critical points. Beginning of the analogy with contact forms: what is a Reeb vector field? Statement of theorem of Hutchings-Taubes on nondegenerate Reeb vector fields with exactly two orbits. The Morse inequalities, part 1: hint of Morse homology. References: Cristofaro-Gardiner note on website (1), Milnor, Sections 4-5.

10/16: Morse inequalities, part 2. Are there Morse functions? What does it mean for a property to be generic? Beginning of the proof that a generic function is Morse. Fredholm maps between Banach spaces. What is a Banach manifold? Statement of Sard-Smale. References: Milnor, Section 5. Hutchings, Section 5.1

10/18: Why is Sard-Smale true? Zeros of equations in two variables which takes values in Banach manifolds. A useful form of Sard-Smale. Proof that generic functions are Morse. What are flow lines? What is the moduli space of flow lines? Definition of Morse Homology. References: Mrowka, Lecture 17. Hutchings, 5.1-5.2.

10/23: Morse homology is isomorphic to singular homology, assuming that the compactification of the space of flow lines is what we expect. Continuation maps. Reference: Hutchings, Section 3

10/25: Proof of the compactification theorem for flow lines, in the once broken flow line case. Reference: Audin-Damian Section 3.2.

10/30: What is a contact manifold? Contact structures, Reeb vector fields. Impressionistic sketch of a contact homology, emphasizing the analogy with Morse theory. J-holomorphic curves. Compatible almost complex structures on symplectizations. Reference: Cristofaro-Gardiner notes on website.

11/1: What kind of problems come up in generalizing the ideas from Morse homology to our setting? “Index difference” makes sense even if the index does not. Transversality problems. Cylinders need not break into cylinders. Statement of the index formula for curves in symplectizations; the Conley-Zehnder index. Reference: Cristofaro-Gardiner notes.

11/6: Morse about the Conley-Zehnder index; computing it in dimension 3. Somewhere injective curves are transverse for generic J. Multiple covers, and the headaches they cause. Different ways to get around these problems; different flavors of contact homology. Reference: Cristofaro-Gardiner notes.

11/8: More about the different flavors of contact homology. The differential on cylindrical contact homology. Hutchings-Nelson paper, part 1. References: Cristofaro-Gardiner notes. Hutchings-Nelson paper at https://arxiv.org/abs/1407.2898

11/13: What kind of breakings can occur on the boundary of the moduli space of cylinders? How does dynamical convexity help us? Relative intersection theory, the relative adjunction formula, and asymptotic analysis. References: Cristofaro-Gardiner notes. Hutchings-Nelson paper at https://arxiv.org/abs/1407.2898

11/15: Proof of d^2 = 0, assuming dynamical convexity. References: Cristofaro-Gardiner notes. Hutchings-Nelson paper at https://arxiv.org/abs/1407.2898

11/20: Morse-Bott theory and computations. There are infinitely many contact structures on the three-torus. References: Cristofaro-Gardiner notes. Hutchings survey at https://math.berkeley.edu/~hutching/pub/tw.pdf, sections 3.3 and 6.6.

11/27: Introduction to embedded contact homology. Definition of the ECH index. The index inequality. Why is the differential well-defined? References: Cristofaro-Gardiner notes. Hutchings survey at https://arxiv.org/pdf/1303.5789.pdf, sections 3.4, 3.5, 5.3.

11/29: More about ECH. Sketch of the proof that d^2 = 0. How would the normal proof of invariance go? Shortcut via Seiberg-Witten theory. References: Cristofaro-Gardiner notes. Hutchings survey at https://arxiv.org/pdf/1303.5789.pdf, sections 5.4 and 5.5.

12/4: Applications of ECH I: from one Reeb orbit to two. References: Cristofaro-Gardiner notes. CG-Hutchings paper at https://arxiv.org/abs/1202.4839.

12/4: Applications of ECH II: symplectic embedding obstructions. References: Cristofaro-Gardiner notes.