A Staircase Proof for Contact Non-Squeezing

Geometry Topology Seminar
Monday, April 1, 2024 - 3:00pm for 1 hour (actually 50 minutes)
Georgia Tech
Lisa Traynor – Bryn Mawr College
Hannah Turner

Gromov's non-squeezing theorem established symplectic rigidity and is widely regarded as one of the most important theorems in symplectic geometry. In contrast, in the contact setting, a standard ball of any radius can be contact embedded into an arbitrarily small neighborhood of a point. Despite this flexibility, Eliashberg, Kim, and Polterovich discovered instances of contact rigidity for pre-quantized balls in $\mathbb R^{2n} \times S^1$ under a more restrictive notion of contact squeezing. In particular, in 2006 they applied holomorphic techniques to show that for any {\it integer} $R \geq 1$, there does not exist a contact squeezing of the pre-quantized ball of capacity $R$ into itself; this result was reproved by Sandon in 2011 as an application of the contact homology groups she defined using the generating family technique. Around 2016, Chiu applied the theory of microlocal sheaves to obtain the stronger result that squeezing is impossible for all $R \geq 1$. Very recently, Fraser, Sandon, and Zhang, gave an alternate proof of Chiu’s nonsqueezing result by developing an equivariant version of Sandon’s generating family contact homology groups. I will explain another proof of Chiu’s nonsqueezing, one that uses a persistence module viewpoint to extract new obstructions from the contact homology groups as defined by Sandon in 2011. This is joint work in progress with Maia Fraser.