Using the geometry of sheaves as the common language, this talk will bridge three separate areas: dynamical systems, signal processing, and data fusion. Because sheaves model consistency relationships between local data, they are easily assembled from detailed models of systems. Being topological in nature, sheaves mediate local-to-global inference. By incorporating local geometry from the start, the global "fit" between local data and models can be quantified, which supports robust inferences about missing or inaccurate data. The utility of this approach is not merely its intellectual cohesion; it also yields performant algorithms. The talk will demonstrate these algorithms on example datasets from music processing and data fusion.