article published yes Rowan Killip author Monica Visan author 056daca0-b8d1-11f0-964f-f91054abf8ca Xiaoyi Zhang author We prove that solutions of the cubic nonlinear Schr\"odinger equation on $\Bbb{R}^2$ can be approximated by a finite-dimensional Hamiltonian system, uniformly on bounded sets of initial data. This is despite the wealth of non-compact symmetries: scaling, translation, and Galilei boosts. Complementing this approximation result, we show that all solutions of the finite-dimensional Hamiltonian system we use can be approximated by the full PDE. A key ingredient in these results is the development of a general methodology for transfering uniform global space-time bounds to suitable Fourier truncations of dispersive PDE models. As an application, we prove symplectic non-squeezing (in the sense of Gromov) for the cubic NLS on $\Bbb{R}^2$. This is the first symplectic non-squeezing result for a Hamiltonian PDE in infinite volume. It is also the first unconditional symplectic non-squeezing result in a scaling-critical setting. Finally, we discuss implications of non-squeezing on the nature of scattering. Johns Hopkins University Press2021 eng 1080-6377 1606.0773810.1353/ajm.2021.0014 1432613-680 yes Killip R, Vişan M, Zhang X. Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2. <i>American Journal of Mathematics</i>. 2021;143(2):613-680. doi:<a href="https://doi.org/10.1353/ajm.2021.0014">10.1353/ajm.2021.0014</a> Killip, Rowan, et al. “Finite-Dimensional Approximation and Non-Squeezing for the Cubic Nonlinear Schrödinger Equation on ℝ2.” <i>American Journal of Mathematics</i>, vol. 143, no. 2, Johns Hopkins University Press, 2021, pp. 613–80, doi:<a href="https://doi.org/10.1353/ajm.2021.0014">10.1353/ajm.2021.0014</a>. R. Killip, M. Vişan, and X. Zhang, “Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2,” <i>American Journal of Mathematics</i>, vol. 143, no. 2. Johns Hopkins University Press, pp. 613–680, 2021. Killip R, Vişan M, Zhang X. 2021. Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2. American Journal of Mathematics. 143(2), 613–680. Killip, Rowan, Monica Vişan, and Xiaoyi Zhang. “Finite-Dimensional Approximation and Non-Squeezing for the Cubic Nonlinear Schrödinger Equation on ℝ2.” <i>American Journal of Mathematics</i>. Johns Hopkins University Press, 2021. <a href="https://doi.org/10.1353/ajm.2021.0014">https://doi.org/10.1353/ajm.2021.0014</a>. R. Killip, M. Vişan, X. Zhang, American Journal of Mathematics 143 (2021) 613–680. Killip, R., Vişan, M., &#38; Zhang, X. (2021). Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2. <i>American Journal of Mathematics</i>. Johns Hopkins University Press. <a href="https://doi.org/10.1353/ajm.2021.0014">https://doi.org/10.1353/ajm.2021.0014</a> 220352026-06-19T07:43:41Z2026-06-22T12:56:27Z