Navigating complex phase diagrams in soft matter systems Colloidal fluids can exhibit complex phase behavior and determining phase diagrams via experiments or computer simulations can be laborious. We demonstrate that the dispersion relation ω(k), obtained from dynamical density functional theory for the uniform density system, is a highly versatile tool for predicting where in the phase diagram complex crystals form. The sign of ω(k) determines whether density modes with wave number k grow or decay over time. We demonstrate the predictive power by investigating the complex phase behavior of particles interacting via core-shoulder pair potentials. With complementary Monte Carlo simulations, we show that regions of the phase diagram where ωðkÞ has one or several unstable (growing) wave numbers are also where crystalline phases occur. Going further, by tuning these unstable wave numbers via the interaction-potential and state-point parameters, we design systems with quasicrystals in the phase diagram. We identify a system with a certain shoulder range exhibiting at least ten different phases. Our general approach accelerates considerably the mapping of complex phase diagrams, crucial for the design of new materials. 136 14 American Physical Society application/pdf