  Most notably we prove that for $d=1,2$ the classical Strichartz norm $$\|e^{i
s\Delta}f\|_{L^{2+4/d}_{s,x}(\mathbb{R}\times\mathbb{R}^d)}$$ associated to the
free Schr\"{o}dinger equation is nondecreasing as the initial datum $f$ evolves
under a certain quadratic heat-flow.
