This lecture covers the theory of mean-field Matsubara dynamics, including Kubo-transformed time correlation functions and exact ring-polymer TCF. It also discusses rovibrational spectroscopy, centroid molecular dynamics, and spurious instantons. The instructor presents the bending of the curve on the curvature problem and the use of normal-mode coordinates. The lecture concludes with a summary of the mean-field Matsubara approach and its application to spectra analysis.