Infinite-dimensional symplectic non-squeezing using non-standard analysis

TL;DR

This paper extends the symplectic non-squeezing theorem to infinite-dimensional Hamiltonian flows by employing non-standard analysis, linking pseudoholomorphic spheres to the maximal existence time of solutions.

Contribution

It introduces a novel approach using non-standard model theory to prove non-squeezing in infinite dimensions without prior knowledge of the theory.

Findings

Established a non-squeezing result for infinite-dimensional Hamiltonian flows.

Connected the maximal existence time to the behavior of pseudoholomorphic curves.

Provided a proof that parallels finite-dimensional Gromov's approach without requiring non-standard analysis expertise.

Abstract

We prove a non-squeezing result for infinite-dimensional Hamiltonian flows using non-standard model theory. For this we prove the existence of a corresponding family of pseudoholomorphic spheres and characterize the maximal time in terms of a limiting behaviour for these curves. While our proof is based on the finite-dimensional results from Gromov's original proof, we do not ask for any prior knowledge of non-standard model theory.