Explore a mathematical lecture that delves into the relationship between scattering diagrams and spectral networks in Calabi-Yau geometries. Learn how Kontsevich-Soibelman and Gross-Siebert developed scattering diagrams for smoothing Calabi-Yau geometries, while spectral networks can be associated with mirror curves of toric Calabi-Yau threefolds. Discover the connection between log BPS numbers, which count maximally tangent curves, and BPS numbers that conjecturally count stable Lagrangians. Examine the correspondence between these BPS numbers for toric Calabi-Yau threefolds without compact divisors through the lens of intermediate quivers, and understand the underlying heuristics of this mathematical relationship.