The study of dynamical systems through a topological lens consistently provides insight and reveals fruitful new approaches. Consciousness, as a phenomenon, is one such -- extraordinarily rich -- dynamical system. Some of its properties, such as the separation between a cognitive system and its environment, raise questions about the spaces in which these systems exist. Other properties, such as the spatial relationships between concepts in conceptual spaces, may allow us to use powerful new tools from topological data analysis such as persistence homology, to examine the global structure of the relationships in a multiscale fashion. Furthermore, by considering mappings between the spaces involved, we may even observe something akin to "particles of thought": topological solitons arising from the dynamics of cognition in non-trivial spaces. In this talk, I will discuss these early-stage ideas along with others we are working on in the biocomputation group at the University of Queensland.