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I am a young postdoc working in symplectic topology. Recently I became intrigued by Floer homotopy, especially after seeing it had been applied to classical questions in symplectic topology. (e.g. Abouzaid and Kragh). This revelation made me excited about the new possibilities that this approach opens up, and I want to try and find other applications.

I am very comfortable with classical Floer theory, both Hamiltonian and Lagrangian, and with constructions such as symplectic homology, TQFT operations, etc. But when it comes to Floer homotopy, my knowledge is pretty much restricted to the slogan, that Floer homotopy cooks up a spectrum from the moduli spaces of Floer solutions, and that this spectrum should see "more information" than mere homology. I am mostly curios about understanding this extra information and how it is utilized to solve problems in Hamiltonian dynamics or topology of Lagrangians.

Which reading (and / or video talks) would you recommend to get a working knowledge of the tools and theory? In a way, I think I am asking the following: Imagine you had a fresh student who wanted to get into the field. What would the reading list you would compile for them contain?

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    $\begingroup$ You may be familiar with these older papers of Ralph Cohen math.stanford.edu/~ralph/morse.ps math.stanford.edu/~ralph/floer.pdf The germs of the ideas are there. Manolescu's work in Seiberg-Witten theory may be closer to what you want. In the symplectic case there are some subtleties that escape me. $\endgroup$
    – Liviu Nicolaescu
    Jun 7, 2020 at 13:26
  • $\begingroup$ Cohen wrote a more up to date review of his take, which also appeared in the handbook of homotopy theory. $\endgroup$
    – Tyrone
    Jun 9, 2020 at 13:03

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