Explore a 30-minute conference talk on constructing a smooth solution to the linear Schrödinger equation on the 2D torus with a vanishing potential. Delve into the adaptation of CKSTT 2010 techniques to demonstrate logarithmic growth of the H^1 Sobolev norm in a linear setting. Examine how nonlinear ideas provide new insights into the linear problem through Fourier mode analysis and a discrete resonant system of ODEs. Discover the construction of a special solution resembling a sequence of finite-dimensional linear oscillators, revealing the energy propagation mechanism to higher frequencies. Gain a deeper understanding of the growth rate control and the underlying growth mechanism in this mathematical exploration presented at the Erwin Schrödinger International Institute for Mathematics and Physics.