Infinite Staircases in Hirzebruch surfaces

Date: 2022-06-28

Time: 16:15 - 17:15

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Dusa McDuff (Online)


The ellipsoidal capacity function $c_X(z)$ of a symplectic four manifold $X$ measures how much the form on $X$ must be dilated in order for it to admit an embedded ellipsoid of eccentricity  $z$.  In most cases there are just finitely many obstructions to such an embedding besides the volume.  If there are infinitely many obstructions, $X$ is said to have a staircase.

This talk will give an almost complete description of these staircases when $X$ is a Hirzebruch surface $H_b$ formed by blowing up  the projective plane with weight $b$.

There is an interweaving, recursively defined, family of obstructions that show there is an open dense set of shape parameters $b$ that are blocked, i.e. have no staircase, and an uncountable number of other values of $b$ that do admit staircases. Moreover, there are interesting symmetries that act on the set of staircases. This is joint work with Nicki Magill and Morgan Weiler.