My primary research interest is the connection between topological dynamical systems and operator algebras. Starting with a dynamical system we construct C*-algebras that are used to ascertain dynamical invariants in a noncommutative framework. Some of my specific interests in this area include hyperbolic dynamical systems called Smale spaces, self-similar group actions, graph and k-graph algebras, and aperiodic substitution tilings. Using the noncommutative geometry program, developed by Alain Connes, we have used these algebras to construct K-theoretic invariants including Poincaré duality classes, KMS equilibrium states, and spectral triples. I have been particularly interested in dynamical systems that are fractal in nature, with comprehensible local structure but chaotic global behaviour. Aperiodic tilings are extremely useful for further understanding this type of phenomenon and serve as a geometric model for self-similar dynamical systems.