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 is a nice introduction to cobordism theory written by T. Weston cobordism notes
- Manuscript on the CW structure coming from a Morse-Smale function.
- The Floer homotopy type of the cotangent bundle