What Operator algebras are abstract objects introduced by Murray and von Neumann a century ago with the intention of providing a rigorous mathematical framework for the theory of quantum mechanics. Since then operator algebras has become an important independent research field which connects to many different aspects of pure mathematics. Topological dynamical systems coming, e.g., from oriented graphs generate a rich class of examples of operator algebras whose structure is reflected in the underlying dynamical system. Using a newly developed reconstruction theory, the aim of this project is to understand and answer long-standing and fundamental questions about such dynamical systems using operator algebraic tools. Why Symbolic dynamics originated at the beginning of the 20th century as a way of modeling a broad class of physical systems with a specified discretized time evolution. However, some of the most fundamental questions, such as the existence of a deciding algorithm for when two symbolic dynamical systems are the same, remain unresolved. Operator algebras provides a new framework from which we can attack these resilient problems. This approach of bridging the two fields is promising as we have already seen a fruitful transportation of ideas over recent years. How As a young researcher it is tremendously beneficial to be around experienced experts engaging in discussion and collaboration. I have spent a year at the University of Wollongong in Australia working with some of the best researchers in the field, and I will spend the next year at the University of Glasgow in Scotland working with a different group of world experts. This provides an excellent environment for me to conduct my research and achieve the goals of the project. SSR This is a project in pure mathematics, it is not motivated by social impact, but aims at nurturing the general knowledge and methods of mathematics itself. Specifically, the project links the fields of topological dynamics and operator algebras.