Office: 383X

Phone: 723-1862

E-mail: rlc@stanford.edu

Class location: TTh 9:00 - 10:20am in 381-T

We will be discussing topics from the following list:

- Recovering the topology of a closed manifold from the flow category of a Morse function and its geometric realization
- The compact moduli spaces of gradient flow trajectories, as framed manifolds with corners
- The Pontrjagin-Thom construction for framed manifolds with corners, and the attaching maps of CW complex
- Constructing cohomology operations from moduli spaces of gradient flow graphs in a manifold, and how they lead to a Topological Field Theory ("Morse Field Theory")
- Morse theory on the loop space of a manifold. Counting geodesics, and constructing String Topology operations Morse theoretically
- Homotopy theoretic aspects of Floer theory, including orientations of moduli spaces of J-holomorphic cylinders.
- The relation between the Floer-Fukaya theory of the cotangent bundle and the string topology of the underlying manifold. This will involve a discussion of the classification of topological field theories coming from the work of Kontsevich and his collaborators, and of Lurie.

Reference Notes and Papers

- Here are the main (rough) lecture notes that I will be using.

- These are M. Hutchings' lecture notes Hutchings notes

- Here is a nice introduction to cobordism theory written by T. Weston cobordism notes

- Manuscript on the classifying space of the flow category. classifyingspace

- Spaces of flow lines and manifolds with corners flows-corners

- Manuscript on the CW structure coming from a Morse-Smale function. CW

- Cohomology operations via Morse theory operations

- Morse Field Theory morse field theory

- Abbondandolo and Schwarz's work on the Floer homology of cotangent bundles Abbondandolo-Schwarz

- Floer's infinite dimensional Morse theory and homotopy theory Floerhomotopy1Floerhomotopy2

- The Floer homotopy type of the cotangent bundle cotangent

- Floer generalized homology theory generalized

- A Morse theoretic view of string topology morse string topology