We review the recent advances on exact results for dynamical correlation functions at large scales and related transport coefficients in interacting integrable models. discuss Drude weights, conductivity diffusion constants, as well linear nonlinear response top of equilibrium non-equilibrium states. consider problems from complementary perspectives general hydrodynamic theory many-body systems...

We produce nontrivial asymptotic estimates for shifted sums of the form ? a ( h ) b m c 2 ? , in which n are un-normalized Fourier coefficients holomorphic cusp forms. These results unconditional, but we demonstrate how to strengthen them under Riemann Hypothesis. As an application, show that there infinitely many three term arithmetic progressions + such ? 0 .