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Floer homology and square pegs


Speaker: Joshua Green, Boston College
Location: 2112 MSB
Start time: Tue, May 7 2024, 2:10PM

The Square Peg Problem asks whether every Jordan curve in the plane inscribes (contains the vertices of) a square.  I will describe a construction in Lagrangian Floer homology based on the inscriptions of a square into a real analytic Jordan curve.  The resulting homology group is very simple -- it is a two-dimensional vector space -- and it has an associated pair of real-valued spectral invariants which encode the "sizes" of some of the squares in the curve.  The spectral invariants can sometimes be used to show that these squares don't shrink out when approximating a rough curve by real analytic ones.   As an application, if a rectifiable Jordan curve encloses an area greater than half that of a circle of equal diameter, then it inscribes a square.   Joint work with Andrew Lobb.