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May 5, 2022

Colloquium- Homotopical methods in Floer theory

The free loop space of a symplectic manifold is equipped with a canonical (multivalued) functional, which assigns to a 1-parameter family of loops the area of the cylinder that they sweep. Floer’s insight that one can assign a homology group to this context by an appropriate reformulation of Morse theory led to a revolution in symplectic topology. Applied to toy examples, Floer’s homology groups agree with ordinary homology. I will discuss an extension of Floer’s idea to generalised cohomology theories. This was first envisioned by Floer himself, but the area of symplectic topology has finally reached the stage where we have concrete applications, which I will describe.