Abstract: The idea that turbulence can be described as a deterministic walk through a repertoire of patterns goes back to Eberhard Hopf. Over the years it was established that these patterns closely correspond to exact coherent structures (ECS) which are often formed by unstable (relative) periodic orbits. Over recent years, a large body of numerical simulations and experiments indicated that the turbulent trajectory moves through the phase space from one ECS to another. The turbulent trajectory approaches the ECS along its stable manifold and leaves along its unstable manifold. It is therefore natural to ask whether these results are coincidental or whether some collection of ECSs can in fact provide a dynamical and statistical description of fluid turbulence. In order to properly answer this question one needs to be able to identify when the turbulent trajectory follows (shadows) a given ECS. However, in systems with continuous symmetries, detecting when the turbulent trajectory shadows a given ECS remains challenging and computationally very expensive. In this talk, we present a novel and computationally efficient approach for detecting the shadowing based on persistent homology. To demonstrate the potential of our method we apply it to the Kuramoto-Sivashinsky equation, which serves as a simple model that mimics some of the properties of fluid turbulence, such as spatiotemporal chaos, in a more accessible setting.