The symplectic non-squeezing theorem, discovered by Gromov in 1985, has been the first result showing a fundamental difference between symplectic transformations and volume preserving ones. A similar but more subtle phenomenon in contact topology was found by Eliashberg, Kim and Polterovich in 2006, and refined by Fraser in 2016 and Chiu in 2017: in this case non-squeezing depends on the size of the domains, and only appears above a certain quantum scale.

In my talk I will outline the geometric ideas behind a proof of this general contact non-squeezing theorem that uses generating functions, a classical method based on finite dimensional Morse theory. This is a joint work with Maia Fraser and Bingyu Zhang.