Explore the mathematical foundations of signal and image analysis in this 53-minute lecture by Ingrid Daubechies from Duke University. Delve into diffusion methods in manifold and fibre bundle learning as part of the AMS Colloquium Lectures at the Joint Mathematics Meetings 2020 in Denver. Discover key concepts such as diffusion distance, Procrustes distance, fiber bundles, and spectral decomposition. Examine applications in biological morphology, data acquisition, and clustering. Gain insights into local and biological representations, geometry, and the importance of interdisciplinary work in advancing mathematical frameworks for signal and image analysis.